Thermodynamic Modeling and Control of Screw Extruder [623419]
Thermodynamic Modeling and Control of Screw Extruder
for 3D Printing
Shumon Koga, David Straub, Mamadou Diagne, and Miroslav Krstic
Abstract — This paper proposes a thermodynamic modeling
and a control design of screw extrusion-based 3D printing of
thermoplastic materials. The model describes the time evolution
of the temperature profile of an extruded polymer by means of a
partial differential equation (PDE) defined on the time-varying
domain. The time evolution of the spatial domain is governed
by an ordinary differential equation (ODE) that reflects the
dynamics of the position of the phase change interface between
polymer granules and molten polymer. Steady-state profile of
the distributed temperature along the extruder is obtained when
the desired setpoint for the interface position is prescribed.
We develop a PDE backstepping state-feedback control law
to stabilize the temperature dynamics at the spatially dis-
tributed steady-state considering a cooling mechanism at the
boundary inlet as an actuator. For some given screw speeds
that correspond to slow and fast operating modes, numerical
simulations are conducted to prove the performance of the
proposed controller. The convergence of the interface position
to the desired setpoint is achieved under physically reasonable
temperature profiles.
I. INTRODUCTION
On the verge of new manufacturing techniques, additive
manufacturing stands out as versatile tool for high flexibility
and fast adaptability in production. It is applicable in a
variety of producing industries, ranging from tissue engi-
neering [16], thermoplastics [21], metal [14] and ceramic
[19] fabrication. One of the most popular types of 3D
printing is Fused Deposition Modeling (FDM) [17], which
uses filaments as raw material, that have to be precisely
manufactured to achieve a good final product quality [1].
From the polymer processing and extrusion cooking indus-
try, screw extruders are well-known devices. Results stated in
[13], [15], [18], [20] give an in-depth description of screw
geometrics, extruder setups and describe the dynamics of
extrusion process consisting of a conveying zone, a melting
zone, and a mixing zone. A mathematical description of such
a model is derived by mass, momentum and energy balances
and appears in coupled transport equations coupled through
a moving interface. This model is used in [15] to describe an
extrusion cooking process. The boundary control of a similar
model is achieved in [3], [4] under the assumption of constant
viscosity.
S. Koga and M. Krstic are with the Department of Mechanical and
Aerospace Engineering, U.C. San Diego, 9500 Gilman Drive, La Jolla, CA,
92093-0411, [anonimizat] [anonimizat]
D. Straub is with the Institute for System Dynamics, University of
Stuttgart, [anonimizat]
M. Diagne is with the Department of Mechanical Aerospace and Nuclear
Engineering of Rensselaer Polytechnic Institute , Troy, NY 12180-3590,
[anonimizat] recent contributions considered screw extrusion as
a useful technology for 3D printing application [2], [5],
[21] allowing to manufacture a wider variety of materials
than FDM, while using polymer granules as raw material
[21]. In [2], a time-delay control was developed on a model
consisting of two phases similarly to [15]. In both cases,
stabilization of the moving interface separating a conveying
and a melting zone is achieved with a fast convergence rate.
Another approach which enables to control screw extruders
in 3D printing is proposed in [5], where an energy-based
model is established, simplifying the implementation of
the control law and circumventing difficulties with state
measurement. In other words, the control of the outflow rate
at the nozzle only relies on the measurement of the heater
current and the screw speed.
In the screw extrusion process, solid material is convected
from the feed to the nozzle located at the end of a heating
chamber. The solid raw material is melted and mixed before
being expelled through the nozzle as a thin filament. For
these process, the thermal behavior is an important factor
which characterize final product quality. In fact, heat is
supplied into the system by the heaters surrounding the
extruder’s barrel on the one hand and by the viscous heat gen-
eration due to a shearing effect [15] on the other hand. The
process of the phase transition from solid to liquid polymer
can be described as a Stefan problem [7]. In this context, the
dynamics of the solid-liquid phase interface is derived from
the energy conservation in which the latent heat required for
melting is driven by the internal heat of the liquid phase,
resulting in the interface velocity to be proportional to the
temperature gradients of the adjacent phase. For instance, in
[6], the Stefan problem for a polymer crystallization process
is described, and the analytical crystallization time is derived.
From a control perspective, the boundary stabilization of
the interface position for a one-phase Stefan problem was
recently developed in [8]–[10] based on the ”backstepping
method” [11], [12]. More precisely, in [9], the observer-
based output feedback control was designed via a nonlinear
backstepping transformation and the exponentially stabiliza-
tion of the closed-loop system was proved without imposing
anya priori assumption. Based on the same technique, in
[10] the state-feedback control design for the one-phase
Stefan problem with flowing liquid was achieved enabling
to exponentially stabilize the system to a constant steady-
state.
In the present paper, we develop the thermodynamic model
of the screw extrusion by describing the phase change as
a two-phase Stefan problem and derive the control law to2018 Annual American Control Conference (ACC)
June 27–29, 2018. Wisconsin Center, Milwaukee, USA
978-1-5386-5428-6/$31.00 ©2018 AACC 2551
melt polymerpolymer granules
faucet
heater
screw
Control of the Tumor Growth Described by FreeBoundary ProblemShumon Koga, Marcella Gomez, and Miroslav KrsticMay 3, 2017Abstract: We consider the tumor growth model described by movingboundary PDE proposed in [1]. Based on our recent contribution in [2], weaim to design the backstepping control law for the model.1 Problem StatementThe tumor growth model proposed by [1] is described by the following cou-pled system on moving boundary:@ @t(r, t)=D1r2@@r✓r2@ @r(r, t)◆+ ( B (r, t)) g1( , ),0<r<R(t).(1)where (r, t) is nutrient concentration of the tumor,D1is the di↵usioncoe cient, Bis a constant nutrient concentration in vasculature (bloodvessel), and is the rate of blood-tissue transfer per unit length (assumedconstant). For the avascular case we have = 0. Assuming that similare↵ects govern the evolution of the inhibitor in the tumor, the followingreaction-di↵usion equation is also obtained@ @t(r, t)=D2r2@@r✓r2@ @r(r, t)◆ g2( , ),0<r<R(t) (2)The dynamics of the moving interface is13s(t)2˙s(t)=Zs(t)0µ ˜˜ r2dr(3)aba1
05010015000.10.20.30.4
Time (min)s(t), ˆs(t)
s(t), stateˆs(t),e s t i m a t i o nsr=0.35mFig. 1. The moving interface.
05010015000.0020.0040.0060.0080.01
Time (min)˜s(t)2
ϵ=0.02ϵ=0.04ϵ=0.06Fig. 2.H1norm of the temperature.VIII. CONCLUSIONS ANDFUTUREWORKSAlong this paper we proposed an observer design andboundary output feedback controller that achieves theexponential stability of sum of the moving interface,H1-norm of the temperature, and estimation error of themthrough a measurement of the moving interface. A nonlinearbackstepping transformation for moving boundary problemis utilized and the controller is proved to keep positive withsome initial conditions, which guarantees some physicalproperties required for the validity of model and the proofof stability. The main contribution of this paper is that,this is the first result which shows the convergence ofestimation error and output feedback systems of one-phaseStefan Problem theoretically. Although the Stefan Problem
0204060801000.30.310.320.330.340.350.36
Time (min)s(t)
Critical region
StateFBOutputFBsr=0.35mFig. 3. The positiveness verification of the controller.has been well known model since 200 years ago relatedwith phase transition which appears in various situationsof nature and engineering, its control or estimation relatedproblem has not been investigated in detail. Towards anapplication to the estimation of sea-ice melting or freezingin Antarctica, it is more practical to construct an observerdesign with a measurement of temperature at one boundary,and it is investigated as a future work.0s(t)LREFERENCES[1]Robert H. Martin and Mark E. Oxley. Moving boundaries in reaction-diffusion systems with absorption.Nonlinear Analysis, 14(2):167 –192, 1990.[2]W. B. Dunbar, N. Petit, P. Rouchon, and Ph. Martin. Motion planningfor a nonlinear stefan problem.ESAIM: Control, Optimisation andCalculus of Variations, 9:275–296, 2003.[3]Bryan Petrus, Joseph Bentsman, and Brian G Thomas. Enthalpy-basedfeedback control algorithms for the stefan problem. InCDC, pages7037–7042, 2012.[4]N. Daraoui, P. Dufour, H. Hammouri, and A. Hottot. Model predictivecontrol during the primary drying stage of lyophilisation.ControlEngineering Practice, 18(5):483–494, 2010.[5]F. Conrad, D. Hilhorst, and T. I. Seidman. Well-posedness of a movingboundary problem arising in a dissolution-growth process.NonlinearAnalysis, 15(5):445 – 465, 1990.[6]A. Armaou and P.D. Christofides. Robust control of parabolic PDEsystems with time-dependent spatial domains.Automatica, 37(1):61 –69, 2001.[7]N. Petit. Control problems for one-dimensional fluids and reactivefluids with moving interfaces. InAdvances in the theory of control,signals and systems with physical modeling, volume 407 ofLecturenotes in control and information sciences, pages 323–337, Lausanne,Dec 2010.[8]Panagiotis D. Christofides. Robust control of parabolic PDE systems.Chemical Engineering Science, 53(16):2949 – 2965, 1998.[9]Bryan Petrus, Joseph Bentsman, and Brian G Thomas. Feedbackcontrol of the two-phase stefan problem, with an application to thecontinuous casting of steel. InDecision and Control (CDC), 201049th IEEE Conference on, pages 1731–1736. IEEE, 2010.[10]Ahmed Maidi and Jean-Pierre Corriou. Boundary geometric control ofa linear stefan problem.Journal of Process Control, 24(6):939–946,2014.[11]C. Karvaris and J. C. Kantor. Geometric methods for nonlinear processcontrol i.Background, Industrial & Engineering Chemistry Research,29:2295–2310, 1990.[12]C Karvaris and J. C. Kantor. Geometric methods for nonlinear processcontrol ii.Controller synthesis, Industrial & Engineering ChemistryResearch, 29:2310–2323, 1990.[13]Ahmed Maidi, Moussa Diaf, and Jean-Pierre Corriou. Boundarygeometric control of a counter-current heat exchanger.Journal ofProcess Control, 19(2):297–313, 2009.[14]Miroslav Krstic and Andrey Smyshlyaev.Boundary control of PDEs:A course on backstepping designs, volume 16. Siam, 2008.[15]A. Smyshlyaev and M. Krstic. Closed-form boundary state feedbacksfor a class of 1-d partial integro-differential equations.AutomaticControl, IEEE Transactions on, 49(12):2185–2202, Dec 2004.[16]Mojtaba Izadi and Stevan Dubljevic. Backstepping output-feedbackcontrol of moving boundary parabolic PDEs.European Journal ofControl, 21(0):27 – 35, 2015.[17]Shuxia Tang and Chengkang Xie. Stabilization for a coupled PDE-ODE control system.Journal of the Franklin Institute, 348(8):2142–2155, 2011.[18]S. Gupta.The classical Stefan problem. Basic concepts, Modellingand Analysis. Applied mathematics and Mechanics. North-Holland,2003.[19]S. Koga, M. Diagne, S. Tang, and M. Krstic. Backstepping control ofa one-phase stefan problem. InACC (accepted), 2016.05010015000.10.20.30.4
Time (min)s(t), ˆs(t)
s(t), stateˆs(t),e s t i m a t i o nsr=0.35mFig. 1. The moving interface.
05010015000.0020.0040.0060.0080.01
Time (min)˜s(t)2
ϵ=0.02ϵ=0.04ϵ=0.06Fig. 2.H1norm of the temperature.VIII. CONCLUSIONS ANDFUTUREWORKSAlong this paper we proposed an observer design andboundary output feedback controller that achieves theexponential stability of sum of the moving interface,H1-norm of the temperature, and estimation error of themthrough a measurement of the moving interface. A nonlinearbackstepping transformation for moving boundary problemis utilized and the controller is proved to keep positive withsome initial conditions, which guarantees some physicalproperties required for the validity of model and the proofof stability. The main contribution of this paper is that,this is the first result which shows the convergence ofestimation error and output feedback systems of one-phaseStefan Problem theoretically. Although the Stefan Problem
0204060801000.30.310.320.330.340.350.36
Time (min)s(t)
Critical region
StateFBOutputFBsr=0.35mFig. 3. The positiveness verification of the controller.has been well known model since 200 years ago relatedwith phase transition which appears in various situationsof nature and engineering, its control or estimation relatedproblem has not been investigated in detail. Towards anapplication to the estimation of sea-ice melting or freezingin Antarctica, it is more practical to construct an observerdesign with a measurement of temperature at one boundary,and it is investigated as a future work.0s(t)LREFERENCES[1]Robert H. Martin and Mark E. Oxley. Moving boundaries in reaction-diffusion systems with absorption.Nonlinear Analysis, 14(2):167 –192, 1990.[2]W. B. Dunbar, N. Petit, P. Rouchon, and Ph. Martin. Motion planningfor a nonlinear stefan problem.ESAIM: Control, Optimisation andCalculus of Variations, 9:275–296, 2003.[3]Bryan Petrus, Joseph Bentsman, and Brian G Thomas. Enthalpy-basedfeedback control algorithms for the stefan problem. InCDC, pages7037–7042, 2012.[4]N. Daraoui, P. Dufour, H. Hammouri, and A. Hottot. Model predictivecontrol during the primary drying stage of lyophilisation.ControlEngineering Practice, 18(5):483–494, 2010.[5]F. Conrad, D. Hilhorst, and T. I. Seidman. Well-posedness of a movingboundary problem arising in a dissolution-growth process.NonlinearAnalysis, 15(5):445 – 465, 1990.[6]A. Armaou and P.D. Christofides. Robust control of parabolic PDEsystems with time-dependent spatial domains.Automatica, 37(1):61 –69, 2001.[7]N. Petit. Control problems for one-dimensional fluids and reactivefluids with moving interfaces. InAdvances in the theory of control,signals and systems with physical modeling, volume 407 ofLecturenotes in control and information sciences, pages 323–337, Lausanne,Dec 2010.[8]Panagiotis D. Christofides. Robust control of parabolic PDE systems.Chemical Engineering Science, 53(16):2949 – 2965, 1998.[9]Bryan Petrus, Joseph Bentsman, and Brian G Thomas. Feedbackcontrol of the two-phase stefan problem, with an application to thecontinuous casting of steel. InDecision and Control (CDC), 201049th IEEE Conference on, pages 1731–1736. IEEE, 2010.[10]Ahmed Maidi and Jean-Pierre Corriou. Boundary geometric control ofa linear stefan problem.Journal of Process Control, 24(6):939–946,2014.[11]C. Karvaris and J. C. Kantor. Geometric methods for nonlinear processcontrol i.Background, Industrial & Engineering Chemistry Research,29:2295–2310, 1990.[12]C Karvaris and J. C. Kantor. Geometric methods for nonlinear processcontrol ii.Controller synthesis, Industrial & Engineering ChemistryResearch, 29:2310–2323, 1990.[13]Ahmed Maidi, Moussa Diaf, and Jean-Pierre Corriou. Boundarygeometric control of a counter-current heat exchanger.Journal ofProcess Control, 19(2):297–313, 2009.[14]Miroslav Krstic and Andrey Smyshlyaev.Boundary control of PDEs:A course on backstepping designs, volume 16. Siam, 2008.[15]A. Smyshlyaev and M. Krstic. Closed-form boundary state feedbacksfor a class of 1-d partial integro-differential equations.AutomaticControl, IEEE Transactions on, 49(12):2185–2202, Dec 2004.[16]Mojtaba Izadi and Stevan Dubljevic. Backstepping output-feedbackcontrol of moving boundary parabolic PDEs.European Journal ofControl, 21(0):27 – 35, 2015.[17]Shuxia Tang and Chengkang Xie. Stabilization for a coupled PDE-ODE control system.Journal of the Franklin Institute, 348(8):2142–2155, 2011.[18]S. Gupta.The classical Stefan problem. Basic concepts, Modellingand Analysis. Applied mathematics and Mechanics. North-Holland,2003.[19]S. Koga, M. Diagne, S. Tang, and M. Krstic. Backstepping control ofa one-phase stefan problem. InACC (accepted), 2016.05010015000.10.20.30.4
Time (min)s(t), ˆs(t)
s(t), stateˆs(t),e s t i m a t i o nsr=0.35mFig. 1. The moving interface.
05010015000.0020.0040.0060.0080.01
Time (min)˜s(t)2
ϵ=0.02ϵ=0.04ϵ=0.06Fig. 2.H1norm of the temperature.VIII. CONCLUSIONS ANDFUTUREWORKSAlong this paper we proposed an observer design andboundary output feedback controller that achieves theexponential stability of sum of the moving interface,H1-norm of the temperature, and estimation error of themthrough a measurement of the moving interface. A nonlinearbackstepping transformation for moving boundary problemis utilized and the controller is proved to keep positive withsome initial conditions, which guarantees some physicalproperties required for the validity of model and the proofof stability. The main contribution of this paper is that,this is the first result which shows the convergence ofestimation error and output feedback systems of one-phaseStefan Problem theoretically. Although the Stefan Problem
0204060801000.30.310.320.330.340.350.36
Time (min)s(t)
Critical region
StateFBOutputFBsr=0.35mFig. 3. The positiveness verification of the controller.has been well known model since 200 years ago relatedwith phase transition which appears in various situationsof nature and engineering, its control or estimation relatedproblem has not been investigated in detail. Towards anapplication to the estimation of sea-ice melting or freezingin Antarctica, it is more practical to construct an observerdesign with a measurement of temperature at one boundary,and it is investigated as a future work.0s(t)LREFERENCES[1]Robert H. Martin and Mark E. Oxley. Moving boundaries in reaction-diffusion systems with absorption.Nonlinear Analysis, 14(2):167 –192, 1990.[2]W. B. Dunbar, N. Petit, P. Rouchon, and Ph. Martin. Motion planningfor a nonlinear stefan problem.ESAIM: Control, Optimisation andCalculus of Variations, 9:275–296, 2003.[3]Bryan Petrus, Joseph Bentsman, and Brian G Thomas. Enthalpy-basedfeedback control algorithms for the stefan problem. InCDC, pages7037–7042, 2012.[4]N. Daraoui, P. Dufour, H. Hammouri, and A. Hottot. Model predictivecontrol during the primary drying stage of lyophilisation.ControlEngineering Practice, 18(5):483–494, 2010.[5]F. Conrad, D. Hilhorst, and T. I. Seidman. Well-posedness of a movingboundary problem arising in a dissolution-growth process.NonlinearAnalysis, 15(5):445 – 465, 1990.[6]A. Armaou and P.D. Christofides. Robust control of parabolic PDEsystems with time-dependent spatial domains.Automatica, 37(1):61 –69, 2001.[7]N. Petit. Control problems for one-dimensional fluids and reactivefluids with moving interfaces. InAdvances in the theory of control,signals and systems with physical modeling, volume 407 ofLecturenotes in control and information sciences, pages 323–337, Lausanne,Dec 2010.[8]Panagiotis D. Christofides. Robust control of parabolic PDE systems.Chemical Engineering Science, 53(16):2949 – 2965, 1998.[9]Bryan Petrus, Joseph Bentsman, and Brian G Thomas. Feedbackcontrol of the two-phase stefan problem, with an application to thecontinuous casting of steel. InDecision and Control (CDC), 201049th IEEE Conference on, pages 1731–1736. IEEE, 2010.[10]Ahmed Maidi and Jean-Pierre Corriou. Boundary geometric control ofa linear stefan problem.Journal of Process Control, 24(6):939–946,2014.[11]C. Karvaris and J. C. Kantor. Geometric methods for nonlinear processcontrol i.Background, Industrial & Engineering Chemistry Research,29:2295–2310, 1990.[12]C Karvaris and J. C. Kantor. Geometric methods for nonlinear processcontrol ii.Controller synthesis, Industrial & Engineering ChemistryResearch, 29:2310–2323, 1990.[13]Ahmed Maidi, Moussa Diaf, and Jean-Pierre Corriou. Boundarygeometric control of a counter-current heat exchanger.Journal ofProcess Control, 19(2):297–313, 2009.[14]Miroslav Krstic and Andrey Smyshlyaev.Boundary control of PDEs:A course on backstepping designs, volume 16. Siam, 2008.[15]A. Smyshlyaev and M. Krstic. Closed-form boundary state feedbacksfor a class of 1-d partial integro-differential equations.AutomaticControl, IEEE Transactions on, 49(12):2185–2202, Dec 2004.[16]Mojtaba Izadi and Stevan Dubljevic. Backstepping output-feedbackcontrol of moving boundary parabolic PDEs.European Journal ofControl, 21(0):27 – 35, 2015.[17]Shuxia Tang and Chengkang Xie. Stabilization for a coupled PDE-ODE control system.Journal of the Franklin Institute, 348(8):2142–2155, 2011.[18]S. Gupta.The classical Stefan problem. Basic concepts, Modellingand Analysis. Applied mathematics and Mechanics. North-Holland,2003.[19]S. Koga, M. Diagne, S. Tang, and M. Krstic. Backstepping control ofa one-phase stefan problem. InACC (accepted), 2016.05010015000.10.20.30.4
Time (min)s(t), ˆs(t)
s(t), stateˆs(t),e s t i m a t i o nsr=0.35mFig. 1. The moving interface.
05010015000.0020.0040.0060.0080.01
Time (min)˜s(t)2
ϵ=0.02ϵ=0.04ϵ=0.06Fig. 2.H1norm of the temperature.VIII. CONCLUSIONS ANDFUTUREWORKSAlong this paper we proposed an observer design andboundary output feedback controller that achieves theexponential stability of sum of the moving interface,H1-norm of the temperature, and estimation error of themthrough a measurement of the moving interface. A nonlinearbackstepping transformation for moving boundary problemis utilized and the controller is proved to keep positive withsome initial conditions, which guarantees some physicalproperties required for the validity of model and the proofof stability. The main contribution of this paper is that,this is the first result which shows the convergence ofestimation error and output feedback systems of one-phaseStefan Problem theoretically. Although the Stefan Problem
0204060801000.30.310.320.330.340.350.36
Time (min)s(t)
Critical region
StateFBOutputFBsr=0.35mFig. 3. The positiveness verification of the controller.has been well known model since 200 years ago relatedwith phase transition which appears in various situationsof nature and engineering, its control or estimation relatedproblem has not been investigated in detail. Towards anapplication to the estimation of sea-ice melting or freezingin Antarctica, it is more practical to construct an observerdesign with a measurement of temperature at one boundary,and it is investigated as a future work.0s(t)LxREFERENCES[1]Robert H. Martin and Mark E. Oxley. Moving boundaries in reaction-diffusion systems with absorption.Nonlinear Analysis, 14(2):167 –192, 1990.[2]W. B. Dunbar, N. Petit, P. Rouchon, and Ph. Martin. Motion planningfor a nonlinear stefan problem.ESAIM: Control, Optimisation andCalculus of Variations, 9:275–296, 2003.[3]Bryan Petrus, Joseph Bentsman, and Brian G Thomas. Enthalpy-basedfeedback control algorithms for the stefan problem. InCDC, pages7037–7042, 2012.[4]N. Daraoui, P. Dufour, H. Hammouri, and A. Hottot. Model predictivecontrol during the primary drying stage of lyophilisation.ControlEngineering Practice, 18(5):483–494, 2010.[5]F. Conrad, D. Hilhorst, and T. I. Seidman. Well-posedness of a movingboundary problem arising in a dissolution-growth process.NonlinearAnalysis, 15(5):445 – 465, 1990.[6]A. Armaou and P.D. Christofides. Robust control of parabolic PDEsystems with time-dependent spatial domains.Automatica, 37(1):61 –69, 2001.[7]N. Petit. Control problems for one-dimensional fluids and reactivefluids with moving interfaces. InAdvances in the theory of control,signals and systems with physical modeling, volume 407 ofLecturenotes in control and information sciences, pages 323–337, Lausanne,Dec 2010.[8]Panagiotis D. Christofides. Robust control of parabolic PDE systems.Chemical Engineering Science, 53(16):2949 – 2965, 1998.[9]Bryan Petrus, Joseph Bentsman, and Brian G Thomas. Feedbackcontrol of the two-phase stefan problem, with an application to thecontinuous casting of steel. InDecision and Control (CDC), 201049th IEEE Conference on, pages 1731–1736. IEEE, 2010.[10]Ahmed Maidi and Jean-Pierre Corriou. Boundary geometric control ofa linear stefan problem.Journal of Process Control, 24(6):939–946,2014.[11]C. Karvaris and J. C. Kantor. Geometric methods for nonlinear processcontrol i.Background, Industrial & Engineering Chemistry Research,29:2295–2310, 1990.[12]C Karvaris and J. C. Kantor. Geometric methods for nonlinear processcontrol ii.Controller synthesis, Industrial & Engineering ChemistryResearch, 29:2310–2323, 1990.[13]Ahmed Maidi, Moussa Diaf, and Jean-Pierre Corriou. Boundarygeometric control of a counter-current heat exchanger.Journal ofProcess Control, 19(2):297–313, 2009.[14]Miroslav Krstic and Andrey Smyshlyaev.Boundary control of PDEs:A course on backstepping designs, volume 16. Siam, 2008.[15]A. Smyshlyaev and M. Krstic. Closed-form boundary state feedbacksfor a class of 1-d partial integro-differential equations.AutomaticControl, IEEE Transactions on, 49(12):2185–2202, Dec 2004.[16]Mojtaba Izadi and Stevan Dubljevic. Backstepping output-feedbackcontrol of moving boundary parabolic PDEs.European Journal ofControl, 21(0):27 – 35, 2015.[17]Shuxia Tang and Chengkang Xie. Stabilization for a coupled PDE-ODE control system.Journal of the Franklin Institute, 348(8):2142–2155, 2011.[18]S. Gupta.The classical Stefan problem. Basic concepts, Modellingand Analysis. Applied mathematics and Mechanics. North-Holland,2003.[19]S. Koga, M. Diagne, S. Tang, and M. Krstic. Backstepping control ofa one-phase stefan problem. InACC (accepted), 2016.
melt polymerpolymer granules
faucet
heater
screw
Control of the Tumor Growth Described by FreeBoundary ProblemShumon Koga, Marcella Gomez, and Miroslav KrsticMay 3, 2017Abstract: We consider the tumor growth model described by movingboundary PDE proposed in [1]. Based on our recent contribution in [2], weaim to design the backstepping control law for the model.1 Problem StatementThe tumor growth model proposed by [1] is described by the following cou-pled system on moving boundary:@ @t(r, t)=D1r2@@r✓r2@ @r(r, t)◆+ ( B (r, t)) g1( , ),0<r<R(t).(1)where (r, t) is nutrient concentration of the tumor,D1is the di usioncoe cient, Bis a constant nutrient concentration in vasculature (bloodvessel), and is the rate of blood-tissue transfer per unit length (assumedconstant). For the avascular case we have = 0. Assuming that similare ects govern the evolution of the inhibitor in the tumor, the followingreaction-di usion equation is also obtained@ @t(r, t)=D2r2@@r✓r2@ @r(r, t)◆ g2( , ),0<r<R(t) (2)The dynamics of the moving interface is13s(t)2˙s(t)=Zs(t)0µ ˜˜ r2dr(3)aba1
05010015000.10.20.30.4
Time (min)s(t), ˆs(t)
s(t), stateˆs(t),e s t i m a t i o nsr=0.35mFig. 1. The moving interface.
05010015000.0020.0040.0060.0080.01
Time (min)˜s(t)2
ϵ=0.02ϵ=0.04ϵ=0.06Fig. 2.H1norm of the temperature.VIII. CONCLUSIONS ANDFUTUREWORKSAlong this paper we proposed an observer design andboundary output feedback controller that achieves theexponential stability of sum of the moving interface,H1-norm of the temperature, and estimation error of themthrough a measurement of the moving interface. A nonlinearbackstepping transformation for moving boundary problemis utilized and the controller is proved to keep positive withsome initial conditions, which guarantees some physicalproperties required for the validity of model and the proofof stability. The main contribution of this paper is that,this is the first result which shows the convergence ofestimation error and output feedback systems of one-phaseStefan Problem theoretically. Although the Stefan Problem
0204060801000.30.310.320.330.340.350.36
Time (min)s(t)
Critical region
StateFBOutputFBsr=0.35mFig. 3. The positiveness verification of the controller.has been well known model since 200 years ago relatedwith phase transition which appears in various situationsof nature and engineering, its control or estimation relatedproblem has not been investigated in detail. Towards anapplication to the estimation of sea-ice melting or freezingin Antarctica, it is more practical to construct an observerdesign with a measurement of temperature at one boundary,and it is investigated as a future work.0s(t)LREFERENCES[1]Robert H. Martin and Mark E. Oxley. Moving boundaries in reaction-diffusion systems with absorption.Nonlinear Analysis, 14(2):167 –192, 1990.[2]W. B. Dunbar, N. Petit, P. Rouchon, and Ph. Martin. Motion planningfor a nonlinear stefan problem.ESAIM: Control, Optimisation andCalculus of Variations, 9:275–296, 2003.[3]Bryan Petrus, Joseph Bentsman, and Brian G Thomas. Enthalpy-basedfeedback control algorithms for the stefan problem. InCDC, pages7037–7042, 2012.[4]N. Daraoui, P. Dufour, H. Hammouri, and A. Hottot. Model predictivecontrol during the primary drying stage of lyophilisation.ControlEngineering Practice, 18(5):483–494, 2010.[5]F. Conrad, D. Hilhorst, and T. I. Seidman. Well-posedness of a movingboundary problem arising in a dissolution-growth process.NonlinearAnalysis, 15(5):445 – 465, 1990.[6]A. Armaou and P.D. Christofides. Robust control of parabolic PDEsystems with time-dependent spatial domains.Automatica, 37(1):61 –69, 2001.[7]N. Petit. Control problems for one-dimensional fluids and reactivefluids with moving interfaces. InAdvances in the theory of control,signals and systems with physical modeling, volume 407 ofLecturenotes in control and information sciences, pages 323–337, Lausanne,Dec 2010.[8]Panagiotis D. Christofides. Robust control of parabolic PDE systems.Chemical Engineering Science, 53(16):2949 – 2965, 1998.[9]Bryan Petrus, Joseph Bentsman, and Brian G Thomas. Feedbackcontrol of the two-phase stefan problem, with an application to thecontinuous casting of steel. InDecision and Control (CDC), 201049th IEEE Conference on, pages 1731–1736. IEEE, 2010.[10]Ahmed Maidi and Jean-Pierre Corriou. Boundary geometric control ofa linear stefan problem.Journal of Process Control, 24(6):939–946,2014.[11]C. Karvaris and J. C. Kantor. Geometric methods for nonlinear processcontrol i.Background, Industrial & Engineering Chemistry Research,29:2295–2310, 1990.[12]C Karvaris and J. C. Kantor. Geometric methods for nonlinear processcontrol ii.Controller synthesis, Industrial & Engineering ChemistryResearch, 29:2310–2323, 1990.[13]Ahmed Maidi, Moussa Diaf, and Jean-Pierre Corriou. Boundarygeometric control of a counter-current heat exchanger.Journal ofProcess Control, 19(2):297–313, 2009.[14]Miroslav Krstic and Andrey Smyshlyaev.Boundary control of PDEs:A course on backstepping designs, volume 16. Siam, 2008.[15]A. Smyshlyaev and M. Krstic. Closed-form boundary state feedbacksfor a class of 1-d partial integro-differential equations.AutomaticControl, IEEE Transactions on, 49(12):2185–2202, Dec 2004.[16]Mojtaba Izadi and Stevan Dubljevic. Backstepping output-feedbackcontrol of moving boundary parabolic PDEs.European Journal ofControl, 21(0):27 – 35, 2015.[17]Shuxia Tang and Chengkang Xie. Stabilization for a coupled PDE-ODE control system.Journal of the Franklin Institute, 348(8):2142–2155, 2011.[18]S. Gupta.The classical Stefan problem. Basic concepts, Modellingand Analysis. Applied mathematics and Mechanics. North-Holland,2003.[19]S. Koga, M. Diagne, S. Tang, and M. Krstic. Backstepping control ofa one-phase stefan problem. InACC (accepted), 2016.05010015000.10.20.30.4
Time (min)s(t), ˆs(t)
s(t), stateˆs(t),e s t i m a t i o nsr=0.35mFig. 1. The moving interface.
05010015000.0020.0040.0060.0080.01
Time (min)˜s(t)2
ϵ=0.02ϵ=0.04ϵ=0.06Fig. 2.H1norm of the temperature.VIII. CONCLUSIONS ANDFUTUREWORKSAlong this paper we proposed an observer design andboundary output feedback controller that achieves theexponential stability of sum of the moving interface,H1-norm of the temperature, and estimation error of themthrough a measurement of the moving interface. A nonlinearbackstepping transformation for moving boundary problemis utilized and the controller is proved to keep positive withsome initial conditions, which guarantees some physicalproperties required for the validity of model and the proofof stability. The main contribution of this paper is that,this is the first result which shows the convergence ofestimation error and output feedback systems of one-phaseStefan Problem theoretically. Although the Stefan Problem
0204060801000.30.310.320.330.340.350.36
Time (min)s(t)
Critical region
StateFBOutputFBsr=0.35mFig. 3. The positiveness verification of the controller.has been well known model since 200 years ago relatedwith phase transition which appears in various situationsof nature and engineering, its control or estimation relatedproblem has not been investigated in detail. Towards anapplication to the estimation of sea-ice melting or freezingin Antarctica, it is more practical to construct an observerdesign with a measurement of temperature at one boundary,and it is investigated as a future work.0s(t)LREFERENCES[1]Robert H. Martin and Mark E. Oxley. Moving boundaries in reaction-diffusion systems with absorption.Nonlinear Analysis, 14(2):167 –192, 1990.[2]W. B. Dunbar, N. Petit, P. Rouchon, and Ph. Martin. Motion planningfor a nonlinear stefan problem.ESAIM: Control, Optimisation andCalculus of Variations, 9:275–296, 2003.[3]Bryan Petrus, Joseph Bentsman, and Brian G Thomas. Enthalpy-basedfeedback control algorithms for the stefan problem. InCDC, pages7037–7042, 2012.[4]N. Daraoui, P. Dufour, H. Hammouri, and A. Hottot. Model predictivecontrol during the primary drying stage of lyophilisation.ControlEngineering Practice, 18(5):483–494, 2010.[5]F. Conrad, D. Hilhorst, and T. I. Seidman. Well-posedness of a movingboundary problem arising in a dissolution-growth process.NonlinearAnalysis, 15(5):445 – 465, 1990.[6]A. Armaou and P.D. Christofides. Robust control of parabolic PDEsystems with time-dependent spatial domains.Automatica, 37(1):61 –69, 2001.[7]N. Petit. Control problems for one-dimensional fluids and reactivefluids with moving interfaces. InAdvances in the theory of control,signals and systems with physical modeling, volume 407 ofLecturenotes in control and information sciences, pages 323–337, Lausanne,Dec 2010.[8]Panagiotis D. Christofides. Robust control of parabolic PDE systems.Chemical Engineering Science, 53(16):2949 – 2965, 1998.[9]Bryan Petrus, Joseph Bentsman, and Brian G Thomas. Feedbackcontrol of the two-phase stefan problem, with an application to thecontinuous casting of steel. InDecision and Control (CDC), 201049th IEEE Conference on, pages 1731–1736. IEEE, 2010.[10]Ahmed Maidi and Jean-Pierre Corriou. Boundary geometric control ofa linear stefan problem.Journal of Process Control, 24(6):939–946,2014.[11]C. Karvaris and J. C. Kantor. Geometric methods for nonlinear processcontrol i.Background, Industrial & Engineering Chemistry Research,29:2295–2310, 1990.[12]C Karvaris and J. C. Kantor. Geometric methods for nonlinear processcontrol ii.Controller synthesis, Industrial & Engineering ChemistryResearch, 29:2310–2323, 1990.[13]Ahmed Maidi, Moussa Diaf, and Jean-Pierre Corriou. Boundarygeometric control of a counter-current heat exchanger.Journal ofProcess Control, 19(2):297–313, 2009.[14]Miroslav Krstic and Andrey Smyshlyaev.Boundary control of PDEs:A course on backstepping designs, volume 16. Siam, 2008.[15]A. Smyshlyaev and M. Krstic. Closed-form boundary state feedbacksfor a class of 1-d partial integro-differential equations.AutomaticControl, IEEE Transactions on, 49(12):2185–2202, Dec 2004.[16]Mojtaba Izadi and Stevan Dubljevic. Backstepping output-feedbackcontrol of moving boundary parabolic PDEs.European Journal ofControl, 21(0):27 – 35, 2015.[17]Shuxia Tang and Chengkang Xie. Stabilization for a coupled PDE-ODE control system.Journal of the Franklin Institute, 348(8):2142–2155, 2011.[18]S. Gupta.The classical Stefan problem. Basic concepts, Modellingand Analysis. Applied mathematics and Mechanics. North-Holland,2003.[19]S. Koga, M. Diagne, S. Tang, and M. Krstic. Backstepping control ofa one-phase stefan problem. InACC (accepted), 2016.05010015000.10.20.30.4
Time (min)s(t), ˆs(t)
s(t), stateˆs(t),e s t i m a t i o nsr=0.35mFig. 1. The moving interface.
05010015000.0020.0040.0060.0080.01
Time (min)˜s(t)2
ϵ=0.02ϵ=0.04ϵ=0.06Fig. 2.H1norm of the temperature.VIII. CONCLUSIONS ANDFUTUREWORKSAlong this paper we proposed an observer design andboundary output feedback controller that achieves theexponential stability of sum of the moving interface,H1-norm of the temperature, and estimation error of themthrough a measurement of the moving interface. A nonlinearbackstepping transformation for moving boundary problemis utilized and the controller is proved to keep positive withsome initial conditions, which guarantees some physicalproperties required for the validity of model and the proofof stability. The main contribution of this paper is that,this is the first result which shows the convergence ofestimation error and output feedback systems of one-phaseStefan Problem theoretically. Although the Stefan Problem
0204060801000.30.310.320.330.340.350.36
Time (min)s(t)
Critical region
StateFBOutputFBsr=0.35mFig. 3. The positiveness verification of the controller.has been well known model since 200 years ago relatedwith phase transition which appears in various situationsof nature and engineering, its control or estimation relatedproblem has not been investigated in detail. Towards anapplication to the estimation of sea-ice melting or freezingin Antarctica, it is more practical to construct an observerdesign with a measurement of temperature at one boundary,and it is investigated as a future work.0s(t)LREFERENCES[1]Robert H. Martin and Mark E. Oxley. Moving boundaries in reaction-diffusion systems with absorption.Nonlinear Analysis, 14(2):167 –192, 1990.[2]W. B. Dunbar, N. Petit, P. Rouchon, and Ph. Martin. Motion planningfor a nonlinear stefan problem.ESAIM: Control, Optimisation andCalculus of Variations, 9:275–296, 2003.[3]Bryan Petrus, Joseph Bentsman, and Brian G Thomas. Enthalpy-basedfeedback control algorithms for the stefan problem. InCDC, pages7037–7042, 2012.[4]N. Daraoui, P. Dufour, H. Hammouri, and A. Hottot. Model predictivecontrol during the primary drying stage of lyophilisation.ControlEngineering Practice, 18(5):483–494, 2010.[5]F. Conrad, D. Hilhorst, and T. I. Seidman. Well-posedness of a movingboundary problem arising in a dissolution-growth process.NonlinearAnalysis, 15(5):445 – 465, 1990.[6]A. Armaou and P.D. Christofides. Robust control of parabolic PDEsystems with time-dependent spatial domains.Automatica, 37(1):61 –69, 2001.[7]N. Petit. Control problems for one-dimensional fluids and reactivefluids with moving interfaces. InAdvances in the theory of control,signals and systems with physical modeling, volume 407 ofLecturenotes in control and information sciences, pages 323–337, Lausanne,Dec 2010.[8]Panagiotis D. Christofides. Robust control of parabolic PDE systems.Chemical Engineering Science, 53(16):2949 – 2965, 1998.[9]Bryan Petrus, Joseph Bentsman, and Brian G Thomas. Feedbackcontrol of the two-phase stefan problem, with an application to thecontinuous casting of steel. InDecision and Control (CDC), 201049th IEEE Conference on, pages 1731–1736. IEEE, 2010.[10]Ahmed Maidi and Jean-Pierre Corriou. Boundary geometric control ofa linear stefan problem.Journal of Process Control, 24(6):939–946,2014.[11]C. Karvaris and J. C. Kantor. Geometric methods for nonlinear processcontrol i.Background, Industrial & Engineering Chemistry Research,29:2295–2310, 1990.[12]C Karvaris and J. C. Kantor. Geometric methods for nonlinear processcontrol ii.Controller synthesis, Industrial & Engineering ChemistryResearch, 29:2310–2323, 1990.[13]Ahmed Maidi, Moussa Diaf, and Jean-Pierre Corriou. Boundarygeometric control of a counter-current heat exchanger.Journal ofProcess Control, 19(2):297–313, 2009.[14]Miroslav Krstic and Andrey Smyshlyaev.Boundary control of PDEs:A course on backstepping designs, volume 16. Siam, 2008.[15]A. Smyshlyaev and M. Krstic. Closed-form boundary state feedbacksfor a class of 1-d partial integro-differential equations.AutomaticControl, IEEE Transactions on, 49(12):2185–2202, Dec 2004.[16]Mojtaba Izadi and Stevan Dubljevic. Backstepping output-feedbackcontrol of moving boundary parabolic PDEs.European Journal ofControl, 21(0):27 – 35, 2015.[17]Shuxia Tang and Chengkang Xie. Stabilization for a coupled PDE-ODE control system.Journal of the Franklin Institute, 348(8):2142–2155, 2011.[18]S. Gupta.The classical Stefan problem. Basic concepts, Modellingand Analysis. Applied mathematics and Mechanics. North-Holland,2003.[19]S. Koga, M. Diagne, S. Tang, and M. Krstic. Backstepping control ofa one-phase stefan problem. InACC (accepted), 2016.05010015000.10.20.30.4
Time (min)s(t), ˆs(t)
s(t), stateˆs(t),e s t i m a t i o nsr=0.35mFig. 1. The moving interface.
05010015000.0020.0040.0060.0080.01
Time (min)˜s(t)2
ϵ=0.02ϵ=0.04ϵ=0.06Fig. 2.H1norm of the temperature.VIII. CONCLUSIONS ANDFUTUREWORKSAlong this paper we proposed an observer design andboundary output feedback controller that achieves theexponential stability of sum of the moving interface,H1-norm of the temperature, and estimation error of themthrough a measurement of the moving interface. A nonlinearbackstepping transformation for moving boundary problemis utilized and the controller is proved to keep positive withsome initial conditions, which guarantees some physicalproperties required for the validity of model and the proofof stability. The main contribution of this paper is that,this is the first result which shows the convergence ofestimation error and output feedback systems of one-phaseStefan Problem theoretically. Although the Stefan Problem
0204060801000.30.310.320.330.340.350.36
Time (min)s(t)
Critical region
StateFBOutputFBsr=0.35mFig. 3. The positiveness verification of the controller.has been well known model since 200 years ago relatedwith phase transition which appears in various situationsof nature and engineering, its control or estimation relatedproblem has not been investigated in detail. Towards anapplication to the estimation of sea-ice melting or freezingin Antarctica, it is more practical to construct an observerdesign with a measurement of temperature at one boundary,and it is investigated as a future work.0s(t)LxREFERENCES[1]Robert H. Martin and Mark E. Oxley. Moving boundaries in reaction-diffusion systems with absorption.Nonlinear Analysis, 14(2):167 –192, 1990.[2]W. B. Dunbar, N. Petit, P. Rouchon, and Ph. Martin. Motion planningfor a nonlinear stefan problem.ESAIM: Control, Optimisation andCalculus of Variations, 9:275–296, 2003.[3]Bryan Petrus, Joseph Bentsman, and Brian G Thomas. Enthalpy-basedfeedback control algorithms for the stefan problem. InCDC, pages7037–7042, 2012.[4]N. Daraoui, P. Dufour, H. Hammouri, and A. Hottot. Model predictivecontrol during the primary drying stage of lyophilisation.ControlEngineering Practice, 18(5):483–494, 2010.[5]F. Conrad, D. Hilhorst, and T. I. Seidman. Well-posedness of a movingboundary problem arising in a dissolution-growth process.NonlinearAnalysis, 15(5):445 – 465, 1990.[6]A. Armaou and P.D. Christofides. Robust control of parabolic PDEsystems with time-dependent spatial domains.Automatica, 37(1):61 –69, 2001.[7]N. Petit. Control problems for one-dimensional fluids and reactivefluids with moving interfaces. InAdvances in the theory of control,signals and systems with physical modeling, volume 407 ofLecturenotes in control and information sciences, pages 323–337, Lausanne,Dec 2010.[8]Panagiotis D. Christofides. Robust control of parabolic PDE systems.Chemical Engineering Science, 53(16):2949 – 2965, 1998.[9]Bryan Petrus, Joseph Bentsman, and Brian G Thomas. Feedbackcontrol of the two-phase stefan problem, with an application to thecontinuous casting of steel. InDecision and Control (CDC), 201049th IEEE Conference on, pages 1731–1736. IEEE, 2010.[10]Ahmed Maidi and Jean-Pierre Corriou. Boundary geometric control ofa linear stefan problem.Journal of Process Control, 24(6):939–946,2014.[11]C. Karvaris and J. C. Kantor. Geometric methods for nonlinear processcontrol i.Background, Industrial & Engineering Chemistry Research,29:2295–2310, 1990.[12]C Karvaris and J. C. Kantor. Geometric methods for nonlinear processcontrol ii.Controller synthesis, Industrial & Engineering ChemistryResearch, 29:2310–2323, 1990.[13]Ahmed Maidi, Moussa Diaf, and Jean-Pierre Corriou. Boundarygeometric control of a counter-current heat exchanger.Journal ofProcess Control, 19(2):297–313, 2009.[14]Miroslav Krstic and Andrey Smyshlyaev.Boundary control of PDEs:A course on backstepping designs, volume 16. Siam, 2008.[15]A. Smyshlyaev and M. Krstic. Closed-form boundary state feedbacksfor a class of 1-d partial integro-differential equations.AutomaticControl, IEEE Transactions on, 49(12):2185–2202, Dec 2004.[16]Mojtaba Izadi and Stevan Dubljevic. Backstepping output-feedbackcontrol of moving boundary parabolic PDEs.European Journal ofControl, 21(0):27 – 35, 2015.[17]Shuxia Tang and Chengkang Xie. Stabilization for a coupled PDE-ODE control system.Journal of the Franklin Institute, 348(8):2142–2155, 2011.[18]S. Gupta.The classical Stefan problem. Basic concepts, Modellingand Analysis. Applied mathematics and Mechanics. North-Holland,2003.[19]S. Koga, M. Diagne, S. Tang, and M. Krstic. Backstepping control ofa one-phase stefan problem. InACC (accepted), 2016.
Fig. 1: Schematic of screw extruder.boundary heat control law to stabilize the interface position atthe desired setpoint is derived, and the stability of the closed-loop system is proved under some realistic assumptions byextending the result in [10]. Finally, simulation results areprovided to illustrate the good performance of the controldesign for some given screw speeds that correspond to slowand fast operating extrusion process.This paper is organized as follows. The thermodynamicmodel of the screw extruder is developed in Section II,and the steady-state analysis is provided in Section III. Thecontrol design is derived in Section IV, and the stability prooffor a specific setup is established in Section V. Simulationresults of polymer extrusion is provided in Section VI witha statement on the control performance. We complete thepaper with our conclusion and future work in Section VII.II. THERMODYNAMICMODEL OFSCREWEXTRUDERWe focus on the thermodynamic model of the screwextrusion process in one-dimensional coordinate along thevertical axis. The model provides the time evolution of thetemperature profile of the extruded material and the interfaceposition between the feeded polymer granules and the moltenpolymer. The granular pellets are conveyed by the screwrotation at a given speedbalong the vertical axis while thebarrel temperature is uniformly maintained atTb. DefiningTs(x, t)andTl(x, t)as the temperature profiles of solid phase(polymer granules) over the spatial domainx2(0,s(t))and liquid phase (molten polymer) over the spatial domainx2(s(t),L), respectively, the following thermodynamicalmodel is derived from the energy conservation and heatconduction laws@Ts@t(x, t)=↵s@2Ts@x2(x, t) b@Ts@x(x, t)+hs(Tb Ts(x, t)),for0<x<s(t),(1)@Tl@t(x, t)=↵l@2Tl@x2(x, t) b@Tl@x(x, t)+hl(Tb Tl(x, t)),fors(t)<x<L .(2)In this paper we consider the temperature distribution in theliquid to be static, and give it in (11) and in Assumption1 at the beginning of Section IV-A. Here,↵i=ki⇢iciandhi=¯hi⇢ici, where⇢i,ci,ki, and¯hifori2{s, l}are thedensity, the heat capacity, the thermal conductivity, and theheat transfer coefficient, respectively and the subscriptssandlare associated to the solid or liquid phase, respectively.The boundary conditions atx=0andx=Lfollow the heatconduction law, and the temperature at the interfacex=s(t)is maintained at the melting pointTm, described as@Ts@x(0,t)= qf(t)ks,Ts(s(t),t)=Tm,(3)@Tl@x(L, t)=q⇤mkl,Tl(s(t),t)=Tm,(4)whereqf(t)<0is a freezing controller at the inlet andq⇤m>0is a heat flux at the nozzle which is assumed to beconstant in time. The interface dynamics is derived by theenergy balance at the interface as⇢s H˙s(t)=ks@Ts@x(s(t),t) kl@Tl@x(s(t),t).(5)The equations (1)-(5) are the solid-liquid phase change modelknown as ”two-phase Stefan problem”.Remark 1:To keep the physical state of each phase, thefollowing conditions must hold:Ts(x, t)Tm,8×2(0,s(t)),8t>0,(6)Tl(x, t) Tm,8×2(s(t),L),8t>0,(7)which represent the model validity conditions.III. STEADY-STATE AND ANALYSISTo ensure a continuous extrusion process, the control ofthe quantity of molten polymer that remains in the extruderchamber at any given time is crucial. By definition, thevolume of fully melted material contained in the chamber isdirectly related to the position of the solid-liquid interfacethat needs to be controlled, consequently. Physically, anygiven position of the interface along the spatial domaincorrespond to a melt temperature profile along the extruder.A. Steady-state solutionAn analytical solution of the steady-state temperatureprofile denoted as(Ts,eq(x),Tl,eq(x))for any given setpointvalue of the interface position defined assr, can be computedby setting the time derivative of the system (1)-(5) to zero.Hence, from (1) and (2) the following set of ordinarydifferential equations in space are obtained(0=↵sT00s,eq(x) bT0s,eq(x)+hs(Tb Ts,eq(x)),0=↵lT00l,eq(x) bT0l,eq(x)+hl(Tb Tl,eq(x)),(8)whereTs,eq(x)2(0,sr)andTl,eq(x)2(sr,L)and theinitial condition are given as(T0s,eq(0) = q⇤fks,Ts,eq(sr)=Tm,T0l,eq(L)=q⇤mkl,Tl,eq(sr)=Tm.(9)
melt polymerpolymer granules
faucet
heater
screw
Control of the Tumor Growth Described by FreeBoundary ProblemShumon Koga, Marcella Gomez, and Miroslav KrsticMay 3, 2017Abstract: We consider the tumor growth model described by movingboundary PDE proposed in [1]. Based on our recent contribution in [2], weaim to design the backstepping control law for the model.1 Problem StatementThe tumor growth model proposed by [1] is described by the following cou-pled system on moving boundary:@ @t(r, t)=D1r2@@r✓r2@ @r(r, t)◆+ ( B (r, t)) g1( , ),0<r<R(t).(1)where (r, t) is nutrient concentration of the tumor,D1is the di usioncoe cient, Bis a constant nutrient concentration in vasculature (bloodvessel), and is the rate of blood-tissue transfer per unit length (assumedconstant). For the avascular case we have = 0. Assuming that similare ects govern the evolution of the inhibitor in the tumor, the followingreaction-di usion equation is also obtained@ @t(r, t)=D2r2@@r✓r2@ @r(r, t)◆ g2( , ),0<r<R(t) (2)The dynamics of the moving interface is13s(t)2˙s(t)=Zs(t)0µ ˜˜ r2dr(3)aba1
05010015000.10.20.30.4
Time (min)s(t), ˆs(t)
s(t), stateˆs(t),e s t i m a t i o nsr=0.35mFig. 1. The moving interface.
05010015000.0020.0040.0060.0080.01
Time (min)˜s(t)2
ϵ=0.02ϵ=0.04ϵ=0.06Fig. 2.H1norm of the temperature.VIII. CONCLUSIONS ANDFUTUREWORKSAlong this paper we proposed an observer design andboundary output feedback controller that achieves theexponential stability of sum of the moving interface,H1-norm of the temperature, and estimation error of themthrough a measurement of the moving interface. A nonlinearbackstepping transformation for moving boundary problemis utilized and the controller is proved to keep positive withsome initial conditions, which guarantees some physicalproperties required for the validity of model and the proofof stability. The main contribution of this paper is that,this is the first result which shows the convergence ofestimation error and output feedback systems of one-phaseStefan Problem theoretically. Although the Stefan Problem
0204060801000.30.310.320.330.340.350.36
Time (min)s(t)
Critical region
StateFBOutputFBsr=0.35mFig. 3. The positiveness verification of the controller.has been well known model since 200 years ago relatedwith phase transition which appears in various situationsof nature and engineering, its control or estimation relatedproblem has not been investigated in detail. Towards anapplication to the estimation of sea-ice melting or freezingin Antarctica, it is more practical to construct an observerdesign with a measurement of temperature at one boundary,and it is investigated as a future work.0s(t)LREFERENCES[1]Robert H. Martin and Mark E. Oxley. Moving boundaries in reaction-diffusion systems with absorption.Nonlinear Analysis, 14(2):167 –192, 1990.[2]W. B. Dunbar, N. Petit, P. Rouchon, and Ph. Martin. Motion planningfor a nonlinear stefan problem.ESAIM: Control, Optimisation andCalculus of Variations, 9:275–296, 2003.[3]Bryan Petrus, Joseph Bentsman, and Brian G Thomas. Enthalpy-basedfeedback control algorithms for the stefan problem. InCDC, pages7037–7042, 2012.[4]N. Daraoui, P. Dufour, H. Hammouri, and A. Hottot. Model predictivecontrol during the primary drying stage of lyophilisation.ControlEngineering Practice, 18(5):483–494, 2010.[5]F. Conrad, D. Hilhorst, and T. I. Seidman. Well-posedness of a movingboundary problem arising in a dissolution-growth process.NonlinearAnalysis, 15(5):445 – 465, 1990.[6]A. Armaou and P.D. Christofides. Robust control of parabolic PDEsystems with time-dependent spatial domains.Automatica, 37(1):61 –69, 2001.[7]N. Petit. Control problems for one-dimensional fluids and reactivefluids with moving interfaces. InAdvances in the theory of control,signals and systems with physical modeling, volume 407 ofLecturenotes in control and information sciences, pages 323–337, Lausanne,Dec 2010.[8]Panagiotis D. Christofides. Robust control of parabolic PDE systems.Chemical Engineering Science, 53(16):2949 – 2965, 1998.[9]Bryan Petrus, Joseph Bentsman, and Brian G Thomas. Feedbackcontrol of the two-phase stefan problem, with an application to thecontinuous casting of steel. InDecision and Control (CDC), 201049th IEEE Conference on, pages 1731–1736. IEEE, 2010.[10]Ahmed Maidi and Jean-Pierre Corriou. Boundary geometric control ofa linear stefan problem.Journal of Process Control, 24(6):939–946,2014.[11]C. Karvaris and J. C. Kantor. Geometric methods for nonlinear processcontrol i.Background, Industrial & Engineering Chemistry Research,29:2295–2310, 1990.[12]C Karvaris and J. C. Kantor. Geometric methods for nonlinear processcontrol ii.Controller synthesis, Industrial & Engineering ChemistryResearch, 29:2310–2323, 1990.[13]Ahmed Maidi, Moussa Diaf, and Jean-Pierre Corriou. Boundarygeometric control of a counter-current heat exchanger.Journal ofProcess Control, 19(2):297–313, 2009.[14]Miroslav Krstic and Andrey Smyshlyaev.Boundary control of PDEs:A course on backstepping designs, volume 16. Siam, 2008.[15]A. Smyshlyaev and M. Krstic. Closed-form boundary state feedbacksfor a class of 1-d partial integro-differential equations.AutomaticControl, IEEE Transactions on, 49(12):2185–2202, Dec 2004.[16]Mojtaba Izadi and Stevan Dubljevic. Backstepping output-feedbackcontrol of moving boundary parabolic PDEs.European Journal ofControl, 21(0):27 – 35, 2015.[17]Shuxia Tang and Chengkang Xie. Stabilization for a coupled PDE-ODE control system.Journal of the Franklin Institute, 348(8):2142–2155, 2011.[18]S. Gupta.The classical Stefan problem. Basic concepts, Modellingand Analysis. Applied mathematics and Mechanics. North-Holland,2003.[19]S. Koga, M. Diagne, S. Tang, and M. Krstic. Backstepping control ofa one-phase stefan problem. InACC (accepted), 2016.05010015000.10.20.30.4
Time (min)s(t), ˆs(t)
s(t), stateˆs(t),e s t i m a t i o nsr=0.35mFig. 1. The moving interface.
05010015000.0020.0040.0060.0080.01
Time (min)˜s(t)2
ϵ=0.02ϵ=0.04ϵ=0.06Fig. 2.H1norm of the temperature.VIII. CONCLUSIONS ANDFUTUREWORKSAlong this paper we proposed an observer design andboundary output feedback controller that achieves theexponential stability of sum of the moving interface,H1-norm of the temperature, and estimation error of themthrough a measurement of the moving interface. A nonlinearbackstepping transformation for moving boundary problemis utilized and the controller is proved to keep positive withsome initial conditions, which guarantees some physicalproperties required for the validity of model and the proofof stability. The main contribution of this paper is that,this is the first result which shows the convergence ofestimation error and output feedback systems of one-phaseStefan Problem theoretically. Although the Stefan Problem
0204060801000.30.310.320.330.340.350.36
Time (min)s(t)
Critical region
StateFBOutputFBsr=0.35mFig. 3. The positiveness verification of the controller.has been well known model since 200 years ago relatedwith phase transition which appears in various situationsof nature and engineering, its control or estimation relatedproblem has not been investigated in detail. Towards anapplication to the estimation of sea-ice melting or freezingin Antarctica, it is more practical to construct an observerdesign with a measurement of temperature at one boundary,and it is investigated as a future work.0s(t)LREFERENCES[1]Robert H. Martin and Mark E. Oxley. Moving boundaries in reaction-diffusion systems with absorption.Nonlinear Analysis, 14(2):167 –192, 1990.[2]W. B. Dunbar, N. Petit, P. Rouchon, and Ph. Martin. Motion planningfor a nonlinear stefan problem.ESAIM: Control, Optimisation andCalculus of Variations, 9:275–296, 2003.[3]Bryan Petrus, Joseph Bentsman, and Brian G Thomas. Enthalpy-basedfeedback control algorithms for the stefan problem. InCDC, pages7037–7042, 2012.[4]N. Daraoui, P. Dufour, H. Hammouri, and A. Hottot. Model predictivecontrol during the primary drying stage of lyophilisation.ControlEngineering Practice, 18(5):483–494, 2010.[5]F. Conrad, D. Hilhorst, and T. I. Seidman. Well-posedness of a movingboundary problem arising in a dissolution-growth process.NonlinearAnalysis, 15(5):445 – 465, 1990.[6]A. Armaou and P.D. Christofides. Robust control of parabolic PDEsystems with time-dependent spatial domains.Automatica, 37(1):61 –69, 2001.[7]N. Petit. Control problems for one-dimensional fluids and reactivefluids with moving interfaces. InAdvances in the theory of control,signals and systems with physical modeling, volume 407 ofLecturenotes in control and information sciences, pages 323–337, Lausanne,Dec 2010.[8]Panagiotis D. Christofides. Robust control of parabolic PDE systems.Chemical Engineering Science, 53(16):2949 – 2965, 1998.[9]Bryan Petrus, Joseph Bentsman, and Brian G Thomas. Feedbackcontrol of the two-phase stefan problem, with an application to thecontinuous casting of steel. InDecision and Control (CDC), 201049th IEEE Conference on, pages 1731–1736. IEEE, 2010.[10]Ahmed Maidi and Jean-Pierre Corriou. Boundary geometric control ofa linear stefan problem.Journal of Process Control, 24(6):939–946,2014.[11]C. Karvaris and J. C. Kantor. Geometric methods for nonlinear processcontrol i.Background, Industrial & Engineering Chemistry Research,29:2295–2310, 1990.[12]C Karvaris and J. C. Kantor. Geometric methods for nonlinear processcontrol ii.Controller synthesis, Industrial & Engineering ChemistryResearch, 29:2310–2323, 1990.[13]Ahmed Maidi, Moussa Diaf, and Jean-Pierre Corriou. Boundarygeometric control of a counter-current heat exchanger.Journal ofProcess Control, 19(2):297–313, 2009.[14]Miroslav Krstic and Andrey Smyshlyaev.Boundary control of PDEs:A course on backstepping designs, volume 16. Siam, 2008.[15]A. Smyshlyaev and M. Krstic. Closed-form boundary state feedbacksfor a class of 1-d partial integro-differential equations.AutomaticControl, IEEE Transactions on, 49(12):2185–2202, Dec 2004.[16]Mojtaba Izadi and Stevan Dubljevic. Backstepping output-feedbackcontrol of moving boundary parabolic PDEs.European Journal ofControl, 21(0):27 – 35, 2015.[17]Shuxia Tang and Chengkang Xie. Stabilization for a coupled PDE-ODE control system.Journal of the Franklin Institute, 348(8):2142–2155, 2011.[18]S. Gupta.The classical Stefan problem. Basic concepts, Modellingand Analysis. Applied mathematics and Mechanics. North-Holland,2003.[19]S. Koga, M. Diagne, S. Tang, and M. Krstic. Backstepping control ofa one-phase stefan problem. InACC (accepted), 2016.05010015000.10.20.30.4
Time (min)s(t), ˆs(t)
s(t), stateˆs(t),e s t i m a t i o nsr=0.35mFig. 1. The moving interface.
05010015000.0020.0040.0060.0080.01
Time (min)˜s(t)2
ϵ=0.02ϵ=0.04ϵ=0.06Fig. 2.H1norm of the temperature.VIII. CONCLUSIONS ANDFUTUREWORKSAlong this paper we proposed an observer design andboundary output feedback controller that achieves theexponential stability of sum of the moving interface,H1-norm of the temperature, and estimation error of themthrough a measurement of the moving interface. A nonlinearbackstepping transformation for moving boundary problemis utilized and the controller is proved to keep positive withsome initial conditions, which guarantees some physicalproperties required for the validity of model and the proofof stability. The main contribution of this paper is that,this is the first result which shows the convergence ofestimation error and output feedback systems of one-phaseStefan Problem theoretically. Although the Stefan Problem
0204060801000.30.310.320.330.340.350.36
Time (min)s(t)
Critical region
StateFBOutputFBsr=0.35mFig. 3. The positiveness verification of the controller.has been well known model since 200 years ago relatedwith phase transition which appears in various situationsof nature and engineering, its control or estimation relatedproblem has not been investigated in detail. Towards anapplication to the estimation of sea-ice melting or freezingin Antarctica, it is more practical to construct an observerdesign with a measurement of temperature at one boundary,and it is investigated as a future work.0s(t)LREFERENCES[1]Robert H. Martin and Mark E. Oxley. Moving boundaries in reaction-diffusion systems with absorption.Nonlinear Analysis, 14(2):167 –192, 1990.[2]W. B. Dunbar, N. Petit, P. Rouchon, and Ph. Martin. Motion planningfor a nonlinear stefan problem.ESAIM: Control, Optimisation andCalculus of Variations, 9:275–296, 2003.[3]Bryan Petrus, Joseph Bentsman, and Brian G Thomas. Enthalpy-basedfeedback control algorithms for the stefan problem. InCDC, pages7037–7042, 2012.[4]N. Daraoui, P. Dufour, H. Hammouri, and A. Hottot. Model predictivecontrol during the primary drying stage of lyophilisation.ControlEngineering Practice, 18(5):483–494, 2010.[5]F. Conrad, D. Hilhorst, and T. I. Seidman. Well-posedness of a movingboundary problem arising in a dissolution-growth process.NonlinearAnalysis, 15(5):445 – 465, 1990.[6]A. Armaou and P.D. Christofides. Robust control of parabolic PDEsystems with time-dependent spatial domains.Automatica, 37(1):61 –69, 2001.[7]N. Petit. Control problems for one-dimensional fluids and reactivefluids with moving interfaces. InAdvances in the theory of control,signals and systems with physical modeling, volume 407 ofLecturenotes in control and information sciences, pages 323–337, Lausanne,Dec 2010.[8]Panagiotis D. Christofides. Robust control of parabolic PDE systems.Chemical Engineering Science, 53(16):2949 – 2965, 1998.[9]Bryan Petrus, Joseph Bentsman, and Brian G Thomas. Feedbackcontrol of the two-phase stefan problem, with an application to thecontinuous casting of steel. InDecision and Control (CDC), 201049th IEEE Conference on, pages 1731–1736. IEEE, 2010.[10]Ahmed Maidi and Jean-Pierre Corriou. Boundary geometric control ofa linear stefan problem.Journal of Process Control, 24(6):939–946,2014.[11]C. Karvaris and J. C. Kantor. Geometric methods for nonlinear processcontrol i.Background, Industrial & Engineering Chemistry Research,29:2295–2310, 1990.[12]C Karvaris and J. C. Kantor. Geometric methods for nonlinear processcontrol ii.Controller synthesis, Industrial & Engineering ChemistryResearch, 29:2310–2323, 1990.[13]Ahmed Maidi, Moussa Diaf, and Jean-Pierre Corriou. Boundarygeometric control of a counter-current heat exchanger.Journal ofProcess Control, 19(2):297–313, 2009.[14]Miroslav Krstic and Andrey Smyshlyaev.Boundary control of PDEs:A course on backstepping designs, volume 16. Siam, 2008.[15]A. Smyshlyaev and M. Krstic. Closed-form boundary state feedbacksfor a class of 1-d partial integro-differential equations.AutomaticControl, IEEE Transactions on, 49(12):2185–2202, Dec 2004.[16]Mojtaba Izadi and Stevan Dubljevic. Backstepping output-feedbackcontrol of moving boundary parabolic PDEs.European Journal ofControl, 21(0):27 – 35, 2015.[17]Shuxia Tang and Chengkang Xie. Stabilization for a coupled PDE-ODE control system.Journal of the Franklin Institute, 348(8):2142–2155, 2011.[18]S. Gupta.The classical Stefan problem. Basic concepts, Modellingand Analysis. Applied mathematics and Mechanics. North-Holland,2003.[19]S. Koga, M. Diagne, S. Tang, and M. Krstic. Backstepping control ofa one-phase stefan problem. InACC (accepted), 2016.05010015000.10.20.30.4
Time (min)s(t), ˆs(t)
s(t), stateˆs(t),e s t i m a t i o nsr=0.35mFig. 1. The moving interface.
05010015000.0020.0040.0060.0080.01
Time (min)˜s(t)2
ϵ=0.02ϵ=0.04ϵ=0.06Fig. 2.H1norm of the temperature.VIII. CONCLUSIONS ANDFUTUREWORKSAlong this paper we proposed an observer design andboundary output feedback controller that achieves theexponential stability of sum of the moving interface,H1-norm of the temperature, and estimation error of themthrough a measurement of the moving interface. A nonlinearbackstepping transformation for moving boundary problemis utilized and the controller is proved to keep positive withsome initial conditions, which guarantees some physicalproperties required for the validity of model and the proofof stability. The main contribution of this paper is that,this is the first result which shows the convergence ofestimation error and output feedback systems of one-phaseStefan Problem theoretically. Although the Stefan Problem
0204060801000.30.310.320.330.340.350.36
Time (min)s(t)
Critical region
StateFBOutputFBsr=0.35mFig. 3. The positiveness verification of the controller.has been well known model since 200 years ago relatedwith phase transition which appears in various situationsof nature and engineering, its control or estimation relatedproblem has not been investigated in detail. Towards anapplication to the estimation of sea-ice melting or freezingin Antarctica, it is more practical to construct an observerdesign with a measurement of temperature at one boundary,and it is investigated as a future work.0s(t)LxREFERENCES[1]Robert H. Martin and Mark E. Oxley. Moving boundaries in reaction-diffusion systems with absorption.Nonlinear Analysis, 14(2):167 –192, 1990.[2]W. B. Dunbar, N. Petit, P. Rouchon, and Ph. Martin. Motion planningfor a nonlinear stefan problem.ESAIM: Control, Optimisation andCalculus of Variations, 9:275–296, 2003.[3]Bryan Petrus, Joseph Bentsman, and Brian G Thomas. Enthalpy-basedfeedback control algorithms for the stefan problem. InCDC, pages7037–7042, 2012.[4]N. Daraoui, P. Dufour, H. Hammouri, and A. Hottot. Model predictivecontrol during the primary drying stage of lyophilisation.ControlEngineering Practice, 18(5):483–494, 2010.[5]F. Conrad, D. Hilhorst, and T. I. Seidman. Well-posedness of a movingboundary problem arising in a dissolution-growth process.NonlinearAnalysis, 15(5):445 – 465, 1990.[6]A. Armaou and P.D. Christofides. Robust control of parabolic PDEsystems with time-dependent spatial domains.Automatica, 37(1):61 –69, 2001.[7]N. Petit. Control problems for one-dimensional fluids and reactivefluids with moving interfaces. InAdvances in the theory of control,signals and systems with physical modeling, volume 407 ofLecturenotes in control and information sciences, pages 323–337, Lausanne,Dec 2010.[8]Panagiotis D. Christofides. Robust control of parabolic PDE systems.Chemical Engineering Science, 53(16):2949 – 2965, 1998.[9]Bryan Petrus, Joseph Bentsman, and Brian G Thomas. Feedbackcontrol of the two-phase stefan problem, with an application to thecontinuous casting of steel. InDecision and Control (CDC), 201049th IEEE Conference on, pages 1731–1736. IEEE, 2010.[10]Ahmed Maidi and Jean-Pierre Corriou. Boundary geometric control ofa linear stefan problem.Journal of Process Control, 24(6):939–946,2014.[11]C. Karvaris and J. C. Kantor. Geometric methods for nonlinear processcontrol i.Background, Industrial & Engineering Chemistry Research,29:2295–2310, 1990.[12]C Karvaris and J. C. Kantor. Geometric methods for nonlinear processcontrol ii.Controller synthesis, Industrial & Engineering ChemistryResearch, 29:2310–2323, 1990.[13]Ahmed Maidi, Moussa Diaf, and Jean-Pierre Corriou. Boundarygeometric control of a counter-current heat exchanger.Journal ofProcess Control, 19(2):297–313, 2009.[14]Miroslav Krstic and Andrey Smyshlyaev.Boundary control of PDEs:A course on backstepping designs, volume 16. Siam, 2008.[15]A. Smyshlyaev and M. Krstic. Closed-form boundary state feedbacksfor a class of 1-d partial integro-differential equations.AutomaticControl, IEEE Transactions on, 49(12):2185–2202, Dec 2004.[16]Mojtaba Izadi and Stevan Dubljevic. Backstepping output-feedbackcontrol of moving boundary parabolic PDEs.European Journal ofControl, 21(0):27 – 35, 2015.[17]Shuxia Tang and Chengkang Xie. Stabilization for a coupled PDE-ODE control system.Journal of the Franklin Institute, 348(8):2142–2155, 2011.[18]S. Gupta.The classical Stefan problem. Basic concepts, Modellingand Analysis. Applied mathematics and Mechanics. North-Holland,2003.[19]S. Koga, M. Diagne, S. Tang, and M. Krstic. Backstepping control ofa one-phase stefan problem. InACC (accepted), 2016.
Fig. 1: Schematic of screw extruder.boundary heat control law to stabilize the interface position atthe desired setpoint is derived, and the stability of the closed-loop system is proved under some realistic assumptions byextending the result in [10]. Finally, simulation results areprovided to illustrate the good performance of the controldesign for some given screw speeds that correspond to slowand fast operating extrusion process.This paper is organized as follows. The thermodynamicmodel of the screw extruder is developed in Section II,and the steady-state analysis is provided in Section III. Thecontrol design is derived in Section IV, and the stability prooffor a specific setup is established in Section V. Simulationresults of polymer extrusion is provided in Section VI witha statement on the control performance. We complete thepaper with our conclusion and future work in Section VII.II. THERMODYNAMICMODEL OFSCREWEXTRUDERWe focus on the thermodynamic model of the screwextrusion process in one-dimensional coordinate along thevertical axis. The model provides the time evolution of thetemperature profile of the extruded material and the interfaceposition between the feeded polymer granules and the moltenpolymer. The granular pellets are conveyed by the screwrotation at a given speedbalong the vertical axis while thebarrel temperature is uniformly maintained atTb. DefiningTs(x, t)andTl(x, t)as the temperature profiles of solid phase(polymer granules) over the spatial domainx2(0,s(t))and liquid phase (molten polymer) over the spatial domainx2(s(t),L), respectively, the following thermodynamicalmodel is derived from the energy conservation and heatconduction laws@Ts@t(x, t)=↵s@2Ts@x2(x, t) b@Ts@x(x, t)+hs(Tb Ts(x, t)),for0<x<s(t),(1)@Tl@t(x, t)=↵l@2Tl@x2(x, t) b@Tl@x(x, t)+hl(Tb Tl(x, t)),fors(t)<x<L .(2)In this paper we consider the temperature distribution in theliquid to be static, and give it in (11) and in Assumption1 at the beginning of Section IV-A. Here,↵i=ki⇢iciandhi=¯hi⇢ici, where⇢i,ci,ki, and¯hifori2{s, l}are thedensity, the heat capacity, the thermal conductivity, and theheat transfer coefficient, respectively and the subscriptssandlare associated to the solid or liquid phase, respectively.The boundary conditions atx=0andx=Lfollow the heatconduction law, and the temperature at the interfacex=s(t)is maintained at the melting pointTm, described as@Ts@x(0,t)= qf(t)ks,Ts(s(t),t)=Tm,(3)@Tl@x(L, t)=q⇤mkl,Tl(s(t),t)=Tm,(4)whereqf(t)<0is a freezing controller at the inlet andq⇤m>0is a heat flux at the nozzle which is assumed to beconstant in time. The interface dynamics is derived by theenergy balance at the interface as⇢s H˙s(t)=ks@Ts@x(s(t),t) kl@Tl@x(s(t),t).(5)The equations (1)-(5) are the solid-liquid phase change modelknown as ”two-phase Stefan problem”.Remark 1:To keep the physical state of each phase, thefollowing conditions must hold:Ts(x, t)Tm,8×2(0,s(t)),8t>0,(6)Tl(x, t) Tm,8×2(s(t),L),8t>0,(7)which represent the model validity conditions.III. STEADY-STATE AND ANALYSISTo ensure a continuous extrusion process, the control ofthe quantity of molten polymer that remains in the extruderchamber at any given time is crucial. By definition, thevolume of fully melted material contained in the chamber isdirectly related to the position of the solid-liquid interfacethat needs to be controlled, consequently. Physically, anygiven position of the interface along the spatial domaincorrespond to a melt temperature profile along the extruder.A. Steady-state solutionAn analytical solution of the steady-state temperatureprofile denoted as(Ts,eq(x),Tl,eq(x))for any given setpointvalue of the interface position defined assr, can be computedby setting the time derivative of the system (1)-(5) to zero.Hence, from (1) and (2) the following set of ordinarydifferential equations in space are obtained(0=↵sT00s,eq(x) bT0s,eq(x)+hs(Tb Ts,eq(x)),0=↵lT00l,eq(x) bT0l,eq(x)+hl(Tb Tl,eq(x)),(8)whereTs,eq(x)2(0,sr)andTl,eq(x)2(sr,L)and theinitial condition are given as(T0s,eq(0) = q⇤fks,Ts,eq(sr)=Tm,T0l,eq(L)=q⇤mkl,Tl,eq(sr)=Tm.(9)Fig. 1: Schematic of screw extruder for original description
(left) and model description (right).
stabilize the interface position assuming a steady-state profile
for the liquid phase. First, we solve the steady-state profile
analytically and derive the condition on the barrel tempera-
ture to a physically consistent operating process. Second, the
boundary heat control law to stabilize the interface position at
the desired setpoint is derived, and the stability of the closed-
loop system is proved under some realistic assumptions by
extending the result in [10]. Finally, simulation results are
provided to illustrate the good performance of the control
design for some given screw speeds that correspond to slow
and fast operating extrusion process.
This paper is organized as follows. The thermodynamic
model of the screw extruder is developed in Section II,
and the steady-state analysis is provided in Section III. The
control design is derived in Section IV, and the stability proof
for a specific setup is established in Section V. Simulation
results are presented in order to analyze the controller’s
performance in Section VI. We complete the paper in Section
VII with concluding remarks and preview further work on
the topic.
II. T HERMODYNAMIC MODEL OF SCREW EXTRUDER
We focus on the thermodynamic model of the screw extru-
sion process in one-dimensional coordinate along the vertical
axis, motivated by [20] which developed a thermodynamic
phase change model for polymer processing. The model
provides the time evolution of the temperature profile of
the extruded material and the interface position between
the feeded polymer granules and the molten polymer. The
granular pellets are conveyed by the screw rotation at a given
speedbalong the vertical axis while the barrel temperature
is uniformly maintained at Tb. DefiningTs(x;t)andTl(x;t)
as the temperature profiles of solid phase (polymer granules)
over the spatial domain x2(0;s(t))and liquid phase
(molten polymer) over the spatial domain x2(s(t);L),
respectively, the following thermodynamical model
@Ts
@t(x;t) = s@2Ts
@x2(x;t) b@Ts
@x(x;t)
+hs(Tb Ts(x;t));for 0<x<s (t);(1)
@Tl
@t(x;t) = l@2Tl
@x2(x;t) b@Tl
@x(x;t)
+hl(Tb Tl(x;t));fors(t)<x<L (2)is derived from the energy conservation and heat conduction
laws. In this paper, we consider the temperature distribution
in the liquid to be static as stated in (11) and in Assumption
1 (see Section IV-A). Here, i=ki
iciandhi=hi
ici,
wherei,ci,ki, and hifori2 fs;lgare the density,
the heat capacity, the thermal conductivity, and the heat
transfer coefficient, respectively and the subscripts sand
lare associated to the solid or liquid phase, respectively.
Referring to [22] which introduces a model of spatially
averaged temperature for screw extrusion, we incorporate the
convective heat transfer through the barrel temperature in (1)
(2). The boundary conditions at x= 0 andx=Lfollow
the heat conduction law, and the temperature at the interface
x=s(t)is maintained at the melting point Tm, described as
@Ts
@x(0;t) = qf(t)
ks; T s(s(t);t) =Tm; (3)
@Tl
@x(L;t) =q
m
kl; T l(s(t);t) =Tm; (4)
whereqf(t)<0is a freezing controller at the inlet and
q
m>0is a heat flux at the nozzle which is assumed to be
constant in time. The interface dynamics is derived by the
energy balance at the interface as
sH_s(t) =ks@Ts
@x(s(t);t) kl@Tl
@x(s(t);t): (5)
The equations (1)-(5) are the solid-liquid phase change model
known as ”two-phase Stefan problem”. Such a phase change
model was developed for polymer processing
Remark 1: In this paper, we assume the pressure in the
chamber to be static and the melting temperature is constant
to avoid supercooling. Then, to keep the physical state of
each phase, the following conditions must hold:
Ts(x;t)Tm;8×2(0;s(t));8t>0; (6)
Tl(x;t)Tm;8×2(s(t);L);8t>0; (7)
which represent the model validity conditions.
III. S TEADY -STATE AND ANALYSIS
To ensure a continuous extrusion process, the control of
the quantity of molten polymer that remains in the extruder
chamber at any given time is crucial. By definition, the
volume of fully melted material contained in the chamber is
directly related to the position of the solid-liquid interface
that needs to be controlled, consequently. Physically, any
given position of the interface along the spatial domain
correspond to a melt temperature profile along the extruder.
A. Steady-state solution
An analytical solution of the steady-state temperature
profile denoted as (Ts;eq(x);Tl;eq(x))for any given setpoint
value of the interface position defined as sr, can be computed
by setting the time derivative of the system (1)-(5) to zero.
Hence, from (1) and (2) the following set of ordinary
differential equations in space are obtained
(
0 = sT00
s;eq(x) bT0
s;eq(x) +hs(Tb Ts;eq(x));
0 = lT00
l;eq(x) bT0
l;eq(x) +hl(Tb Tl;eq(x));(8)2552
and the boundary values are given as
(
T0
s;eq(0) = q
f
ks; T s;eq(sr) =Tm;
T0
l;eq(L) =q
m
kl; T l;eq(sr) =Tm:(9)
At equilibrium, the interface equation (5) satisfies the fol-
lowing equality
0 =ksT0
s;eq(sr) klT0
l;eq(sr): (10)
The solution to the set of differential equations (8) has the
following form
(
Tl;eq(x) =p1eq1(x sr)+p2eq2(x sr)+Tb;
Ts;eq(x) =p3eq3(x sr)+p4eq4(x sr)+Tb;(11)
where
q1=b+pb2+ 4 lhl
2 l; q 2=b pb2+ 4 lhl
2 l;(12)
q3=b+pb2+ 4 shs
2 s; q 4=b pb2+ 4 shs
2 s:(13)
Letr=Tb Tm. Substituting (11) into the boundary
conditions (9) and (10), we obtain
p1=rq2eq2(L sr)+q
m=kl
q1eq1(L sr) q2eq2(L sr); (14)
p2= rq1eq1(L sr)+q
m=kl
q1eq1(L sr) q2eq2(L sr); (15)
p3=rq4+K=k s
q3 q4; p 4= rq3 K=k s
q3 q4; (16)
K=klr( q1q2)
eq1(L sr) eq2(L sr)
+ (q1 q2)q
m
q1eq1(L sr) q2eq2(L sr);
(17)
and the steady-state input is given by
q
f=p3q3e q3sr+p4q4e q4sr: (18)
Hence, once the parameters (sr;Tb;q
m)are prescribed, the
steady-state input is uniquely obtained.
B. Barrel temperature condition for a valid steady-state
For the model validity, the steady-state must satisfy (6) and
(7) which restricts the barrel temperature to some physically
admissible values.
Lemma 1: If the barrel temperature satisfies
qTb Tmq; (19)
where
q=(q1 q2)q
m
qden;q= q
m
klq2eq2(L sr); (20)
qden= klq1q2
eq1(L sr) eq2(L sr)
+ksq3
q1eq1(L sr) q2eq2(L sr)
; (21)
then the steady-state solution satisfies (6) and (7).
Proof: SinceTl;eq(sr) =Tm, it is necessary to have
T0
l;eq(sr)0which yields
p1q1+p2q20: (22)Substituting (14) and (15) into (22), we get
Tb Tm(q1 q2)q
m
klq1q2
eq1(L sr) eq2(L sr); (23)
knowing that q1q2<0. With the help of (22) and
from (11) the derivative of Tl;eq(x)satisfiesT0
l;eq(x)
p1q1
eq1(x sr) eq2(x sr)
. Thus, the sufficient condition
ofT0
l;eq(x)0for8x2(sr;L)isp1q10which yields
Tb Tm q
m
klq2eq2(L sr): (24)
Next, the solid steady-state satisfies Ts;eq(sr) =Tm, so it is
necessary to have T0
s;eq(sr)0leading top3q3+p4q4
0which trivially holds under condition of (22). Hence,
from (11), the derivative of Ts;eq(x)satisfiesT0
s;eq(x)
p4q4
eq3(x sr)+eq4(x sr)
. Then, the sufficient condition
forT0
s;eq(x)0isp4q40, which yields
Tb Tm (q1 q2)q
m
qden: (25)
One can notice that condition (25) is less conservative than
condition (23). Hence, combining (24) and (25), we conclude
Lemma 1.
IV. C ONTROL DESIGN OF BOUNDARY HEAT
When the solid pellets are injected and heated into the
extruder chamber, the amount of the molten polymer expands
reducing the quantity of solid material into the chamber.
Thus a cooling effect arising from the continuous feeding
of cooler pellets enables to maintain the interface at the de-
sired setpoint. The setpoint open-loop boundary heat control
qf(t) =q
f(see (9)) is not sufficient to drive the solid-liquid
interface position to the desired setpoint. In this section, we
develop the control design of the boundary heat at the inlet
to drive the interface to the setpoint while stabilizing the
temperature profile at the steady-state.
A. Reference error system for a dynamics reduced to a single
phase
First, we impose the following assumption on the liquid
temperature.
Assumption 1: The liquid temperature is at steady-state
profile, i.e.Tl(x;t) =Tl;eq(x).
Let defineu(x;t)andX(t)as the reference error variables
of solid phase temperature evolution, namely, u(x;t) =
ks(Ts;eq(x) Ts(x;t)); X (t) =s(t) sr. With the help
of (3), we get
u(s(t);t) =ks(Ts;eq(s(t)) Tm): (26)
In addition, rewriting (5) in term of u(x;t)leads to the
following equation of interface dynamics
_X(t) = @u
@x(s(t);t) +
ksT0
s;eq(s(t)) klT0
l;eq(s(t))
:
(27)
Taking a linearization of the right hand side of (26) and (27)
with respect to s(t)around the setpoint srand by the steady2553
state solutions in (11), the dynamics of the reference error
system is obtained as
@u
@t(x;t) = s@2u
@x2(x;t) b@u
@x(x;t) hsu(x;t);(28)
@u
@x(0;t) = U(t); (29)
u(s(t);t) =CX(t); (30)
_X(t) =AX(t) @u
@x(s(t);t); (31)
where = (sH) 1,U(t) = (qf(t) q
f), and
C=ks(p3q3+p4q4); (32)
A=
ks(p3q2
3+p4q2
4) kl(p1q2
1+p2q2
2)
: (33)
B. Backstepping transformation
Consider the bacsktepping transformation
w(x;t) =u(x;t)
sZs(t)
x (x y)u(y;t)dy
(x s(t))X(t); (34)
which maps to
@w
@t(x;t) = s@2w
@x2(x;t) b@w
@x(x;t) hsw(x;t)
+ _s(t) 0(x s(t))X(t);0<x<s (t)(35)
@w
@x(0;t) =
w(0;t); (36)
w(s(t);t) =CX(t); (37)
_X(t) = (A c)X(t) @w
@x(s(t);t); (38)
where
=b
2 sandc >0is a gain parameter. Taking the
derivatives of (34) in xandtalong the solution of (28)-
(31), in order to satisfy (35), (37), and (38), the gain kernel
function is uniquely given by
(x) =c
(d1 d2)
ed1x ed2x
; (39)
whered1,d2are
d1=b+p
D
2 s; d 2=b p
D
2 s: (40)
withb=b+ CandD=b2+4 s
A b
sC+hs
. Finally,
using (36), the controller is given by
U(t) =
u(0;t)
sZs(t)
0f(x)u(x;t)dx
f(s(t))X(t); (41)
where
f(x) =c
(d1 d2)
(d1
)e d1x (d2
)e d2x
:
(42)
Rewriting (41) with respect to qf(t)andTs, we have
qf(t) =q
f
ks(Ts(0;t) Ts;eq(0))
ks
sZs(t)
0f(x)(Ts(x;t) Ts;eq(x))dx
+f(s(t))(s(t) sr): (43)V. T HEORETICAL ANALYSIS FOR A SPECIFIC SETUP
While the controller is designed through the backstepping
method, the stability of the target system is not proven
theoretically. Moreover, the condition of model validity needs
to be satisfied under the control law. To achieve a theoretical
result, in this section we impose a following assumption.
Assumption 2: The barrel temperature is set as melting
temperature and the external heat input is zero, i.e.
Tb=Tm; q
m= 0; (44)
Corollary 1: Under Assumption 2, the steady state pro-
files (11), and steady state input (18) becomes Tl;eq(x) =
Tm,Ts;eq(x) =Tm, andq
f= 0. Also,C= 0 andA= 0.
In addition, the following setpoint restriction is given.
Assumption 3: The setpoint is chosen to satisfy
sr>s0+ ks
sZs0
0f(x)
f(s0)(Tm Ts;0(x))dx; (45)
wheres0=s(0)andTs;0(x) =Ts(x;0).
The main theorem is stated as following.
Theorem 1: Under Assumption 2, the closed-loop system
consisting of the plant (28)-(31) and the control law (41)
satisfies (6) and (7) and is exponentially stable in the sense
ofH1norm.
The proof of Theorem 1 is established by showing that
(6) and (7) are satisfied and employing a Lyapunov analysis.
A. Model validity condition
LetZ(t)be defined as
Z(t) =U(t) +
u(0;t)
=
sZs(t)
0f(x)u(x;t)dx f(s(t))X(t):(46)
The following lemma is stated.
Lemma 2: The following properties hold:
Z(t)>0; (47)
u(x;t)>0;_s(t)>0; (48)
s(0)<s(t)<sr: (49)
Proof: Taking the time derivative of (46) along the
solution of (28)–(31), we have
_Z(t) = cZ(t) _s(t)f0(s(t))X(t): (50)
Assume that (47) is not valid, which implies 9t>0such
that
Z(t)>0;8t2(0;t); Z (t) = 0: (51)
Similarly to [7], by Maximum principle and Hopf’s lemma,
we getu(x;t)>0and_s(t)>0for allx2(0;s(t))and
t2(0;t). Thus, we have s(t)> s 0>0,8t2(0;t).
In addition, using (46) and knowing that f(x)>0, it
leads toX(t)<0,8t2(0;t). Therefore, (50) leads to
_Z(t)> cZ(t);8t2(0;t). Gronwall’s inequality leads
to the inequality regarding the solution of the differential
equation, written as Z(t)Z(0)e ct;8t2(0;t]. Thus,
we haveZ(t)Z(0)e ct>0, which contradicts with the
assumption (51). Hence, (47) is proved. Then, by Maximum
principle, (48) holds. Imposing (47) and (48) on (46), we
obtainX(t)<0which leads to (49).2554
B. Stability analysis
Introducez(x;t) =w(x;t)e
x. Then, by (35)-(38), z-
system is given by
@z
@t(x;t) = s@2z
@x2(x;t) z(x;t)
+ _s(t) 0(x s(t))e
xX(t); (52)
@z
@x(0;t) =0; z(s(t);t) = 0; (53)
_X(t) = cX(t) e
s(t)@z
@x(s(t);t); (54)
where:=b2
4 s+hs. Consider the following functional
V=1
2jjzjj2
H1+p
2X(t)2: (55)
Taking the time derivative of (55) along the solution of (52)-
(54), we have
_V= sjjzxxjj2 ( s+)jjzxjj2 jjzjj2
+ _s(t)Zs(t)
0 0(x s(t))e
xz(x;t)dxX(t)
_s(t)zx(s(t);t)2
2+zx(s(t);t) 0(0)e
s(t)X(t)
Zs(t)
0f(x)zx(x;t)dxX(t)!
pcX(t)2 p zx(s(t);t)e
s(t)X(t): (56)
Choosep=c
2 2sre2
sr. Applying Young’s, Cauchy
Schwartz, Poincare’s, and Agmon’s inequalities to (56) with
the help of (48) and (49), we obtain
_V s
4s2r+ 1+
jjzjj2
H1 pc
2X(t)2
+ _s(t)
M1jjzjj2+M2jjzxjj2+ 2c2
2+ 2
X(t)2
;
(57)
whereM1=Rsr
0 0( x)2dxandM2=Rsr
0(f0( x))2dx.
Hence, (57) can be written as
_V dV+a_s(t)V; (58)
whered > 0anda > 0are defined as d=
2 minn
s
4s2r+1+;co
,a= maxn
M1;M2;2
p
c2
2+ 2o
.
Following the procedure in [10], the inequality (58) with
(48) and (49) leads to the exponential norm estimate
V(t)ea(s(t) s0)V(0)e dteasrV(0)e dt: (59)
Let (t) =jjwjj2
H1+X(t)2. Then, we have MV
(t)MV where M= 2 maxfe2
sr(1 +
2);1
pg,
M =
maxf2(1 +
2);p
2g 1. Therefore, (t)
M
Measr (0)e dt, which proves the exponential stability of
the targetw-system inH1-norm. Since the u-system in (28)-
(31) and the target w-system in (35)-(38) have equivalent
stability property due to the invertibility of the backstepping
transformation (34), the exponential estimate in H1-norm is
also guaranted for the u-system, which concludes the proof
of Theorem 1.TABLE I: HDPE parameters obtained by [20].
melting point Tm 135C
specific heat solid cs 1895 Jkg 1K 1
specific heat melt cl 2640 Jkg 1K 1
therm. conduct. solid ks 0:373 Wm 1K 1
therm. conduct. melt kl 0:324 Wm 1K 1
solid density s 955 kgm 3
melt density l 780 kgm 3
heat of fusion H 39000 Jkg 1
VI. S IMULATION
For simulative investigation of the controller’s perfor-
mance in different operating conditions, we implemented
the solid phase model (1), (3), (5) with the controller (43)
assuming that the liquid phase is at the steady-state Tl;eq(x).
We used the boundary immobilization method to obtain a
fixed boundary system and discretized the system with finite
differnces to construct a finite dimensional representation
of the model. Using Matlab’s ode23s solver, we simulated
the setup with three different advection speeds ranging from
2 mm/s to 50 mm/s, to cover a wide spectrum of operating
modes. The material parameters are chosen from [20], in
which distinct values for high density polyethylene in solid
and liquid state were experimentally derived (see Table I).
The extruder length is L= 10 cm and the constant barrel
temperature is set to Tb= 145C. Further, the auxil-
iary heat input was chosen q
m= 100 W=m2. The initial
temperature is set as a linear profile, which satisfies the
boundary condition (4). For each advection speed, the control
parametercis adjusted to generate a reasonable boundary
temperature. While higher advection speeds enable shorter
convergence times, it is only achieved when the control
gaincis also sufficiently large. However, larger values in
the control gain will initially produce very low boundary
temperature values. For each configuration, we adjusted the
control gain cto produce comparable temperature peaks
at the inlet and prevent unrealistic values. The closed-loop
responses of the interface position s(t), the boundary control
inputqf(t), and the boundary temperature Ts(0;t)are shown
in Fig. 2, Fig. 3 and Fig. 4, respectively. The interface
responses, depicted in Fig. 2, have quite similar behaviors in
all three setups. However, the control input, shown in Fig. 3,
appears to act faster for higher advection speeds but exhibits
a similar qualitative behavior. Comparable observations were
made with the input temperature response (Fig.4). Note that
Fig. 2 and Fig. 3 have different time ranges.
For verification of our established control law, we also
tested a closed loop setup with linear control laws. We found,
that our method delivers a superior performance, as the linear
control is not able to handle the strongly nonlinear behavior
of the plant. The PI-control was implemented as
qf(t) =q
f+KP(s(t) sr) +KIZt
t0(s( ) sr)d : (60)
and could not achieve convergence to the setpoint sr, but only
slowly decaying oscillation centered at an offset value. With2555
0 3 6 9 12 1554321
time (min)position (cm)b= 2 mm =s
b= 10 mm =s
b= 50 mm =s
Fig. 2: Interface position for different advecions speeds.
0 0.5 1 1.5 2 2.5 3 1 0:50105
time (s)control
W=m2
b= 2 mm =s
b= 10 mm =s
b= 50 mm =s
Fig. 3: Inlet heat control qf(t)in short time scale. Z(t)>0
is observed for each screw speed.
0 10 20 30 40 50 602580135
time (s)boundary temp (C)
b= 2 mm =s
b= 10 mm =s
b= 50 mm =s
Fig. 4: Inlet temperature in short time scale.
that response, the validity condition (7) cannot be ensured.
From the simulations, we conclude that our control design
achieves a stable interface position, even with very fast
advection speeds 50 mm/s, with which a particle inserted in
the inlet will travel in two seconds through the extruder,
when assuming a 10 cm extruder.
VII. C ONCLUSIONS AND FUTURE WORK
In this paper, we developed the thermodynamic model and
control design for the screw extrusion based 3D printing.
The steady-state analysis is provided, and the control design
to stabilize the interface position is derived. The simulation
results prove the effectiveness of the boundary feedback
control law for some given screw speeds. For further inves-
tigation, the observer-based output feedback design utilizing
the measurements of the temperature at the inlet and the
interface position will be considered as a future work.
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