Extrusion Process Control: Modeling, Identification, and ... · Extrusion Process Control:...

134 IEEE TRANSACTIONS ON CONTROL SYSTEMS TECHNOLOGY, VOL. 6, NO. 2, MARCH 1998

Extrusion Process Control: Modeling,Identification, and Optimization

Brian R. Tibbetts,Member, IEEE,and John Ting-Yung Wen,Senior Member, IEEE

Abstract—Our work has been focused on developing a method-ology for process control of bulk deformation—specifically, extru-sion. A process model is required for control. We state the difficul-ties which exist with currently available models. An appropriatedecomposition and associated assumptions are introduced whichpermit a real-time process model to be constructed via spectralapproximation to solve the axi-symmetric two-dimensional (2-D) transient heat conduction equation with heat generation andloss. This solution, simulation results, and model execution timesare provided. We use this model development plus output re-lationships from previous model development work to developparameter identification and open-loop control methodologies.These methodologies are motivated by the structure and proper-ties of the developed model. We demonstrate that the parametersand control variables enter into the model equations so that theidentification and open-loop optimization problems are tractable.An example with plant trial data is provided.

Index Terms—Approximation methods, identification, metalindustry, modeling, optimization methods, partial differentialequations, process control, spectral approximation.

I. THE PROCESS

T HE direct extrusion process is illustrated in Fig. 1. Theram (1) applies force to thedummy block(2), transmitting

the force to thebillet (5). Initially, this force causes upset—thewhole billet is deformed to fit thecontainer (3) shape andthere is initial flow from thedie (4). After upset, the extrusionprocess takes place, producing the desiredextrudate(6). Themetal for the extrudate comes from thedeformation zone(5a) within the billet. The material within thedead metalzone(5b) remains within the container throughout the process.During the entire process, the material flow and shape of thedeformation and dead metal zones are evolving. After most ofthe billet has been extruded, the flow contorts to such a degreethat extrusion is no longer feasible. The remainder of the billetis removed and the entire process is repeated.

The range of extruded products is typically large; therefore,the control objectives vary greatly from product to product.

Manuscript received January 1997; revised October 1997. This work wassupported by the Aluminum Processing Program at Rensselaer PolytechnicInstitute, a consortium of ten Aluminum Extruders, a U.S. Departmentof Energy Fellowship for Integrated Manufacturing, and the New YorkState Center for Advanced Technology (CAT) in Automation, Robotics, andManufacturing. The CAT is partially funded by a block grant from the NewYork State Science and Technology Foundation.

B. R. Tibbetts is with the New York State Center for Advanced Technologyin Automation, Robotics, and Manufacturing, Rensselaer Polytechnic Institute,Troy, NY 12180-3590 USA.

J. T.-Y. Wen is with the Electrical, Computer, and Systems EngineeringDepartment, Rensselaer Polytechnic Institute Troy, NY 12180-3590 USA.

Publisher Item Identifier S 1063-6536(98)01819-3.

Fig. 1. Schematic of direct extrusion.

These objectives may include, for example, surface quality,microstructure uniformity, and production efficiency. Somespecific examples are:

• stock shapes for secondary processing—production effi-ciency is very important since these items have a com-modity pricing structure;

• furniture components—surface quality is key;• architectural components—surface quality and mi-

crostructural properties are both critical since strengthand aesthetics are required.

The goal of our work is to develop cost-effective strategiesfor control of this process. These strategies need to address awide range of product geometries, processing conditions, qual-ities requirements, and productivity goals. A process model isrequired to achieve this goal.

The process model enters explicitly into an identifier orobserver, operating in real-time plus it may be directly in-corporated into a feedback controller. Therefore, this modelmust be designed to capture the response of very differentsystems (for example, round rod with area reduction of tentimes or rectangular tubing with area reduction of 500 times)and require modest computational power.

II. BACKGROUND

Process control of bulk deformation has received relativelylittle attention from the control systems community. Muchof the work found has been limited by the existing models.For example, Meyer and Wadley [3] developed a model-based feedback control for material properties and inequalityconstraints for part yielding. This approach could only beapplied to relatively slow processes due to model complexity;in this particular case, the hot isostatic process required 2 hto perform.

Previously reported work has focused on problems wheresimplifications are appropriate which are not broadly applica-

1063–6536/98$10.00 1998 IEEE

TIBBETTS AND WEN: EXTRUSION PROCESS CONTROL 135

ble. For example, there has been an ongoing effort in control ofisothermal thermal forging. The researchers involved includeJ. Adams, S. S. Banda, J. Berg, R. Chang, A. Chaudhary, R.V. Grandhi, D. Irwin, A. Kumar, J. C. Malas, III, M. Mears,C. A. Schwartz, and V. Seetharaman. Most of this body ofwork [4]–[10] assumes that there is no temperature changein the workpiece during the process. The condition does notapply to metal extrusion and is not expected to be applicableto many other deformation processes such as drawing. Also,it is notable that even with the above the simplifications, theresulting models were not suitable for real-time control.

There has been some work which is based on removingthe need for a model. The previously cited group proposedusing artificial neural networks to emulation the complexfinite element method (FEM)-based model and open-loopoptimization [7], [9], [10]. This effort did not remove theconstant temperature restriction or faced the prospect of com-putationally expensive network retraining. Another exampleis cyclic learning control for extrusion developed by Panditand Buchheit [11], [12]. This work employed a black-boxmodel and required strong assumptions on process linearity,actuation behavior, and constancy of disturbances. Many typ-ical deformation processes will violate one or more of theseassumptions.

Based on our above assessment of the available work andits possible application to extrusion, an existing model wassought. These models could be classified into two broad areas:FEM-based models (detailed) or averaged-variable models(coarse, approximation-based). The FEM-based models pro-vide the needed information, but cannot be implemented inreal time [13]–[15]. The averaged-variable models are com-putationally tractable; however, they do not provide localizedinformation needed for control [16], [17]. Therefore, we choseto develop a model of the extrusion process specifically forreal-time control.

III. T HE MODEL FOR CONTROL DESIGN

The thermo-mechanical response of the billet during ex-trusion is governed by a set of nonlinear, coupled partialdifferential equations given by 1) heat transfer and 2) defor-mation approximation

(1)

s.t. (2)

These equations are coupled via the flow stress of the materialwhich is part of the heat generation expression andthe deformation power .

Our purpose requires that these equations be solved; how-ever, direct solution cannot be done in real time. The followingassumptions are introduced.

1) The deformation field determined with constant flowstress will closely approximate the optimum field in theset of possible fields.

Fig. 2. Thermo-mechanical model components.

2) The set of possible fields can be restricted to includeonly fields which are linear in ram velocity.

3) Model errors introduced by using nonlocal flow stressare acceptable.

Assumption 1) accomplishes the decoupling of (1) and (2) bynot considering the temperature dependence of the materialflow stress. Assumption 2) allows the model description to beindependent of the control, the ram velocity. Assumptions 2)and 3) permit the most intense calculations to be done off-line.

The decomposition of the thermo-mechanical model isshown in Fig. 2. Development of the heat transfer and de-formation components are described below. The heat transfermodel component establishes the extrusion model frameworkor basic model structure. The deformation model compo-nent, in conjunction with the structure of the heat transfermodel, provides the unique attributes for a given extrusion.The material response can be included directly by using thetemperature predicted by the heat transfer model and thestrain rate predicted by the deformation model to determinean appropriate flow stress value. This flow stress value, inturn, influence the subsequent temperature distributions.

Our model is based on an axi-symmetric extrusion, roundbillet to round extrudate. This restriction has been chosen sinceeven with the simplifications introduced thus far, there is littlehope of a real-time solution for an arbitrary shape. We expectthat the analytical model will capture a large portion of theresponse for an arbitrary extrusion shape and the black-boxmodel will capture the remainder.

This problem is naturally described in cylindrical coordi-nates. The -axis is the centerline through the center of thedie port and the billet. The-axis extends radially from the

-axis. The origin is at the die entrance. The current billetlength (positive direction) is denoted by . The coordinatesystem, main geometric parameters, and the flow boundaryfunction are illustrated in Fig. 3.

The remainder of this section will be focused on solving (1).This solution will rely on the linear dependence of the flowvelocity field from the ram velocity; however, no specific formwill be assumed. The next section will review the basis for (2)and the development of one possible family of flow velocitieswhich satisfy this equation.

A. State Equation Development

There are two physical quantities from which the states andtheir evolution equations are developed: the billet size and thebillet temperature distribution. The system input is the ram


Fig. 3. Problem coordinates.

speed, . The billet size state can be represented by thebillet length, , with the evolution equation

(3)

Development of the states and evolution equation for the bil-let temperature distribution begins with the governing equationfor heat transfer. The governing equation is

(4)

with the following boundary conditions:

where is the thermal diffusivity, is the density ofaluminum, is the specific heat capacity, is temperature

is the heat generation term, and is represents thebillet boundary.

The changing boundary and the loss of mass via theextrudate adds an interesting component to this problem.We have transformed this equation so that one can applyfinite-difference or spectral approximation to obtain a solutionaccording to the following rules:

The governing equation becomes

(5)

where the flow velocity is represented as

We have chosen spectral approximation for the solution;the partial differential equation is transformed into an infinite-dimension system of ordinary differential equations. An appro-priate set of basis functions (Bessel functions inand cosines

in ) allows the desired boundary conditions to be imposed.The temperature is approximated by

(6)

based on the following eigenvalue problems.Eigenvalue Problem #1:

Solution:

where

zeroth order Bessel function, first kind;first-order Bessel function, first kind;zeros of .

Inner Product:

Eigenvalue Problem #2:

Solution:

whereInner Product:

The original equation, (5), is transformed from a partialdifferential equation to a finite system of ordinary differentialequations. Equation (5) is projected onto the basis set of thetemperature approximation using the inner product operationsdefined previously. Integration by parts is used to exchangehigher order derivatives of the temperature field for higherorder derivatives of the basis vectors. The stabilizing influenceof the conduction term permits the infinite dimensional systemof ODE’s to be truncated.

Integration by parts of the conduction term also requiresthat the temperature gradient boundary conditions be incorpo-rated into the solution. Gradient boundary conditions whichconsist of temperature-dependent and constant terms can beincorporated.

After the above manipulation and rearrangement into amatrix form, the final state equations become

(7)


where

......

......

is the “projected temperature state”, is coefficientmatrix for the conduction term, is the coefficient matrixfor the advective term (rate of change in temperature due tomotion of a bulk with a temperature gradient), is thecoefficient matrix for the heat generation term, is theboundary condition contribution, is the ram velocity, and

is the ram position.This system of equations is in a very desirable form for the

control purpose and has the following properties.

• Temperature evolution is nonlinear with respect to theram position (measurable).

• Temperature evolution is bilinear with respect to thetemperature state, , and the control, .

• There are free coefficients in and whichmay be determined by through identification.

• The elements of and are determined solelyby problem geometry and the inner product operation forthe spectral approximation.

• The elements of and are determined by thedeformation component , and and thespectral approximation.

• The stabilizing, conduction term dominates as higherorder basis functions are included in the approximation,permitting the infinite-dimension system of equations tobe truncated to a very low order.

B. Boundary Condition Definition

The model given in Section III-A is for the generalizedboundary condition, . In this section, themodel is made more specific by chosing the gradients tobe independent of temperature. Three gradient functions arespecified at 1) the container wall; 2) the die face; and 3) the

dummy block

The heat transfer boundary term is derived from threefunctions, , , and , which characterize theboundary. The following decomposition is introduced so thatspecific parameters are available for identification and theparameters enter into the equation linearly

Using the above decomposition, the boundary conditionterm in the temperature state equation can be ex-pressed as

(8)

where

The variable represents the configuration adjustable com-ponent of the boundary conditions. For example, the simu-lation results shown subsequently assume perfect insulatingconditions at the extrudate and constant leaky conditions atthe die. The coefficients represent the magnitude of thesefunctions.

C. Heat Generation Term Definition

A similar decomposition can be performed on the heatgeneration term . This term is constructed from deforma-tion, friction, and shearing sources; the following equationillustrates this calculation:

(9)

The heat generation functions have the following dependen-cies:

where , and represent the scaling of theassumed flow stress. Therefore, the heat generation term can


be represented as

(10)

The specific heat generation functions, , , and, will be developed in Section IV.

D. Output Equations

There are many possible output relations which one canderive from this model. For our purposes, we will present threeparticular equations, the extrudate temperature, the extrusionload, and the maximum strain rate within the billet.

1) Extrudate Temperature:The output equation for the ex-trudate temperature depends greatly on the method of measure-ment or on the use of the variable. Generically, this variablecan be calculated as follows:

If one wishes to consider the process performance or a researchsetup which uses an instrumented die with a thermal couple,then the coefficients, , are time-invariant and canbe written as . The typical measurementcapability in the production environment requires a morecomplicated description.

The production environment may use either a pyrometer oran infrared camera for determining the extrudate temperature,if this temperature is measured at all. Either of these deviceswill measure temperature on the extrudate where the extrudateexits the die bolster (a substantial support structure whichholds the die in place). This point is approximately 3 to 4 ftfrom the die exit. A very thorough description would requirethat one consider the continued conduction throughout theextrudate, losses to the environment, and the movement ofthe extrudate from the die exit to the point of measurement.A reasonable approximation is to consider the following.

1) There is no heat loss to the environment.2) There is no heat conduction axially.3) The heat completely equalizes radially due to the high

conductivity of aluminum.

Thus, the coefficients must contain a time delay,dependent on , and the projection of the billetmodes at the die face onto the zeroth radial mode of theextrudate

2) Extrusion Load: The same phenomena which generatesheat also requires the press exert a force on the billet to causethe billet to be extruded. Generically, the relationship can be

represented as follows:

(11)

The specific load functions, , , and , will bedeveloped in Section IV. A bias termhas also been includedto account for various losses such as friction in the press anderrors in load measurement.

3) Maximum Strain Rate:Our investigation into the phe-nomenological effects of the extrusion process on the qualityof the extrudate show that the strain rate and, in particular,the local strain is one indicator of quality (see Shepard [18]).This investigation did not, however, indicate precisely how thislocal strain rate should be defined so that one has a parameterwhich is a good indicator of product quality. We, therefore,suggest that the maximum strain rate within the billet be usedas an approximation to this indicator.

By construction of the flow field description (Section IV),this parameter is linear in

(12)

where represents the functional description of the strainrate.

E. Model Summary

The model is summarized in (13)

(13a)

(13b)

(13c)

(13d)

(13e)

where

“projected temperature state;”ram position;ram velocity;conduction;boundary heat loss;advection (rate of change in temperature due tomotion of a bulk with a temperature gradient);heat generation;extrudate temperature;output matrix for ;extrusion load;maximum strain rate;map, linear in ram velocity and nonlinear in ramposition and billet location, to strain rate.


IV. FLOW FIELD DEVELOPMENT

We have fully developed the deformation model componentin earlier work [1], [2]. A brief summary of these results isprovided for completeness.

We choseupper bound analysis(UBA) as the basis of thedeformation model component. UBA is the search for thekinematically admissible flow field which requires theminimum power

s.t.

According to the UBA theorem [19], this flow field is theactual flow field. Conventional applications of UBA werespecifically for determining required process power. Typically,simple flow fields have been used which predict the power;however, they bare little resemblance to the physical flowfields observed experimentally. This is not suitable for ourpurpose.

Han et al. [20] developed a flow field approximation foruse in UBA for a contoured die, very light reduction extrusionproblem. This particular flow field did appear to be a closerapproximation to the physical situation than the typical UBAflow field. The flow field includes a bulk component (fixed)and a variational component. The parameters of the variationalcomponent of the flow field are chosen to minimize the processpower via a numerical optimization algorithm.

We have made the following improvements in the originalwork for application to typical industrial extrusion controlproblems.

• Determined required conditions on the flow boundary forapplication to flat-faced dies, , shown in Fig. 3.

• Generalized the structure of the variational component ofthe flow field.

• Determined required conditions on the flow field de-scription so that it is linear in ram velocity (the controlvariable).

• Implemented a hybrid optimization algorithm for deter-mining the free parameters in the flow field descrip-tion. The algorithm includes line searches, modified linesearches, and steepest descent methods. Increases in speedand robustness against local minima are observed.

With the conditions specified, the optimization can be per-formed off-line, independent of the control.

The conventional UBA applied to flat-face dies requiredthat the material ultimately going into the die (region 5ain Fig. 1) be artificially divided into an undeformed region(far from the die) and a deformation zone. This division isdifficult to specify and generates local discontinuities. Thesedifficulties are avoided with our approach. We demonstratedgood qualitative prediction of experimental measurements ofa flow field with our deformation model.

The kinematically admissible velocity field description isconstructed by forming the axial component,, first andthen directly calculating the radial velocity component.The first term (bulk component) in the axial velocity functionguarantees that the same amount material flows through allaxial cross sections. The remaining terms with constraints areconstructed to locally modify the flow via polynomial basic

functions without affecting the bulk material flow. This resultis given below

(14)

subject to the following constraints:

where

The strain rate tensor components are defined as

and the strain rate magnitude, , is defined as

The power optimization is described by (15) where isthe minimum power resulting from a flow field described in theprevious section with the most favorable set of free parametersand boundary description,

(15)

There are three contributing factors to the power used duringextrusion: power directly required to deform the material,

, power require to overcome friction at the container wall,, and power required to shear the material in the dead

zone-deformation zone interface, . All of these termsdepend on the velocity of the flow field and, therefore, on the


free parameters introduced above. The power components are

(16)

where is the height of the dead zone, , , andare the same scaling factors as in (10), andis the materialflow stress.

By the development of the velocity field, the ram velocity, enters into the velocity field and strain rate parameters

linearly

From (17), the load terms are defined as

The velocity field, via the definitions for the extrusion load,is used to construct the heat generation terms which resultfrom deformation, internal shear, and friction. The followingterms are used in (9):

The heat generation kernel due to deformation is the integrandfrom the extrusion load analog. The heat generation kernelsdue to friction and shear are slightly more complicated sincethey are not defined throughout the volume. The introductionof the Dirac delta function and application of appropriatescaling permit these function to be used in the spectralapproximation.

V. IDENTIFICATION

The model description in (13) is physically motivated;therefore, parameters for identification should be quantities

which are not specifically defined by the modeling effort. Theconduction and advective terms in the temperature state equa-tion (13a) are completely defined by the modeling process;the conduction term is solely a function of the billet geometryand the advective term is defined by the flow field. Theheat boundary condition describes the gradient at theboundary and is not specified. The heat generation term

depends on 1) the flow field (fixed by constructionof the deformation model component) and 2) the material flowstress and coefficients of friction and shear. The coefficients offriction and shear are not specified and, therefore, suitable foridentification. The flow stress is also suitable for identificationsince it appears as a constant. This constant represents anaggregate of the constitutive relationship operating on the flowfield and the temperature distribution.

The ram position state transition equation (13b) does nothave parameters for identification. Similarly, the temperatureoutput equation (13c) does not have parameters requiringidentification since this equation depends only on the projectedtemperature state and the workpiece geometry.

The extrusion load equation does have the same parametersfor identification that the heat generation term has. The loadequation can be defined in terms of the heat generation term

(17)

The parameters for identification are expressed for conve-nience as

A. Estimated Extrudate Temperature

Identification will use measured temperature, , and itsestimate, . This measurement is discrete withsamples.Therefore, and

The estimate for measured temperature is calculated by usingthe forth-order Runge–Kutta integration algorithm to transformthe continuous state equation (13a) into a discrete representa-tion

(18)

where is the output matrix,and are the number of


basis functions in and , respectively, and

...

......

where

Also, note that represents the conduction matrix at theth instantance . Similarly, represents the

advection matrix at theth instantance .It is important to note that (18) is linear in the parameters to

be identified, . This linearity will be maintained as long as:

• flow stress is not treated as an explicit function oftemperature (an assumption in the model development);

• boundary temperature gradients are not temperature de-pendent.

The linearity of this relationship greatly improves the solvabil-ity of the problem over a typical nonlinear description.

Equation (18) is linear in the initial state, , also. Thisproperty will prove very useful in solving the open-loopcontrol problem discussed in Section VI-C.

B. Estimated Extrusion Load

The estimated extrusion load is given by

(19)

VI. RESULTS

In this section, we first present simulation results for themodel, including predicted internal billet temperature distri-bution. Next, our initial parameter identification technique forthis model is shown. Finally, using the equations developed forthe identification, we solve an open-loop control problem. Ex-perimental data from an experimental extrusion press and froma production press is used for comparison and identification.

A. Model

The temperature predictions shown in Fig. 4 and in Fig. 5are based on experimental extrusion of AA6060 by Lefstad[21]. The billet is 16.5-cm length and 10.0-cm diameter. Theextrudate is 1.04-cm diameter and the ram velocity throughoutthe push is 2.0 cm/s. Experimentally measured extrudatetemperature is shown in Fig. 4.

Fig. 4. Predicted extrudate temperature.

The model-generated results were produced in the followingmanner.

1) Constructed a flow field for the extrusion geometry andtabulated by current billet length (off-line).

2) Calculate and which depend on the flowfield (off-line).

3) Calculate and , (off-line).4) Solve the temperature evolution equation (on-line).

These results reasonably approximate the experimental mea-surements for most of the extrusion. Also, the model predictsthat the core will be cooler than the extrudate surface, consis-tent with available information.

The isothermal plots for different times during the pushshow that the majority of the temperature increase is generatedby the intense shearing between the deformation zone and deadzone, regions 5a and 5b in Fig. 1, respectively. Note, the heavyboundary represents the current billet shape and the extrudatesize (toward the bottom).

The off-line calculations for the simulation shown required2 h with a 586, 100 MHz running Windows NT 3.51. Theresults shown required 1.5 min to run the 7.0-s extrusion.These results are for a 64-state model (88 basis functions).Many of the features predicted by this model precision arepresent in a 16-state model (4 4 basis functions). Thereduced-state model for this example executes in 3.0 s on thesame platform. Although some care was exercised in codedevelopment to optimize for speed, much more can be donefor a real-time implementation. Based on number of operationsand raw processor performance, improvements up to a factorof ten might be realizable on this platform. In addition, low-cost hardware alternatives exist which might realize similarimprovements. Therefore, the high-resolution model can beimplemented in real-time.

B. Parameter Identification

The identification problem will use both measured tem-perature and extrusion load. The problem will be cast as


Fig. 5. Predicted temperature history of billet during extrusion. Note: Thinner lines are used to represent lower temperatures.

TABLE IPARAMETER IDENTIFICATION: PARAMETER CONSTRAINTS

a constrained quadratic optimization problem. Earlier effortsused a least squares approach which did generate parameters;however, these parameters were not physically realizable. Theidentification problem requires that the following minimizationbe solved [22]:

(20)

where is the set of permissible parameter values and thecost function is defined as

(21)

where represents the weighting on the measured temper-ature error and is the weighting on the extrusion loaderror.

The weighting parameters have two purposes. First, they canbe used to reflective relative confidence in the measurement ac-curacy. Second, the force and temperature measurement havedissimilar units and different magnitudes. These differencesimply a weighting factor, perhaps an inappropriate weighting.Explicit inclusion of a weighting factor and an expectation of

TABLE IIPARAMETER IDENTIFICATION: PARAMETER VALUES

the magnitudes of the different data items permits adjustmentof the weighting parameters for a balanced cost function.

The nonlinear programming algorithm,constr() in MAT-LAB was used with the constraints shown in Table I tominimize the cost function. The weightings in the cost functionwere set with temperature measured in degrees Celius andforce in kiloNewtons: and .

The resulting parameter values are provided in Table II.The plant data used for the identification and the result modelpredictions are shown in Fig. 6. The comparison of actual dataand model prediction with errors generally below 10%. A trendis noted in the temperature prediction being positively biasedand the load being negatively biased.

The identified parameters were used in the model with a newplant trial data set to gage the success of the identificationprocedure. The extrusion which generated this new data setused a different initial billet temperature profile and a differentram speed trajectory. Fig. 7 shows the experimental data andthe model predictions. The prediction of trends and generalagreement are good. Although still within 10%, the error intemperature prediction has increased compared with the errorpresent in the identification run. The error in load predictionhas decreased.


Fig. 6. Comparison of data and model prediction after identification.

Fig. 7. Model predictions for the reference extrusion.

Figs. 8 and 9 illustrate the differences in ram speed trajec-tory and initial billet temperature profile between the extrusionused for identification and the extrusion used as reference tocheck the identification.

Both the identification run and the verification run showtemperature predictions above the measured temperature. Webelieve that there are two sources of this discrepancy. First,improvements in the deformation model for short billets arerequire. In general, the force predictions for short billets aretoo high which will result in elevated temperature prediction.Second, constant flow stress is not accurate and is expectedto have a substantial impact on the temperature prediction.The validation run started with a billet which was 40 to 70Chotter than the billet used for identification. Also, the extrudatetemperature was 20C hotter. Therefore, the flow stress of thisbillet will substantially lower than the identification billet, thusproducing less heat during extrusion. To see the effects, theverification run was rerun with a 15% reduction in the flowstress parameter. The results are shown in Fig. 10. We see that

Fig. 8. Comparison between identification and reference extrusions: Ramspeed trajectory.

Fig. 9. Comparison between identification and reference extrusions: Initialbillet temperature profile.

Fig. 10. Model predictions for the reference extrusion with reduced flowstress.

this validation run has significantly less error than the originalvalidation run.

One source of difficulty for using this data set for identifica-tion is the lack of variation in the input signal (the ram speed,shown in Fig. 8). It is highly likely that this signal does notsatisfy the persistency of excitation condition, resulting in onlypartial identification, rather than unique determination, of theparameters.

C. Open-Loop Control

The range of extruded products and their applications arelarge. As such, the applicable measures of quality and theirrequired levels vary from product to product. These require-ments can be mapped to specific values of the physical processvariables [9], [18], [5]. We have chosen constant maximum


Fig. 11. Comparison of the optimized and reference ram speed and maxi-mum strain rate trajectories.

Fig. 12. Predicted extrudate temperature using optimized ram speed trajec-tory and initial billet temperature. The reference extrudate temperature is alsoshown.

strain rate and constant extrudate temperature to representgood quality for this example.

The maximum strain rate, given by (13e), is linear in the ramspeed, ; however, it is nonlinear in the billet length. Also, itis not dependent on the projected temperature state variables.The temperature evolution equation (13a) depends on boththe initial projected temperature state and the ram speedtrajectory . Therefore, the suggested design procedure isto determine the desired ram speed trajectory which resultsin the appropriate maximum strain rate. With the ram speeddetermined, one can then find an initial billet temperatureprofile which achieve a constant extrudate temperature for theram speed trajectory.

The required ram speed trajectory can be calculated easilysince the strain rate output equation (13e) is a static map.Therefore, the desired ram speed trajectory is given by

(22)

The choice of desired maximum strain rate must still be made.For this particular example, we adjusted the value so that ouroptimized extrusion finished approximately 10% faster thanthe extrusion shown in the previous section for identificationvalidation. The ram speed profiles and the associated maxi-mum strain rate predictions for the referenced plant trial andthe open-loop control design are shown in Fig. 11.

Fig. 13. Comparison of the optimized and reference initial billet temperatureprofiles.

TABLE IIIOPTIMIZATION: INITIAL BILLET TEMPERATURE MODE CONSTRAINTS AND VALUES

To determine the desired initial billet temperature profile,we start by recalling the equation for the estimated extrudatetemperature (18)

(23)

Note, we have expressed the implicit dependence of theand on the ram speed trajectory. Since ram speed

is known, all parts of (23) are known except for . Anapproach analogous to the parameter identification problem fordetermining is motivated since enters into (23) linearly.

Finding the best initial billet temperature profile requiresthat the following minimization be solved:

(24)

where is the set of permissible initial projected temperaturestates and the cost function is defined as

(25)

Exactly as was done for the identification problem, the non-linear programming algorithm,constr()in MATLAB was usedwith the constraints shown in Table III to minimize the costfunction, . The constraints on the initial billet temperaturewere driven by the actuation capabilities. We assumed thatthe best one could hope is that a “reasonably smooth” profilecould be imposed axially and that there is no radial variability.The first six states represent the constant radial variable axialbasis functions. Since the greater index represents a mode withhigher spatial frequency, these modes were subject to tighterrestrictions to impose the “reasonably smooth” criteria.

The control problem was solved with desired extrudatetemperature being arbitrarily specified at 520C. The requiredinitial projected temperature modes are given in Table III.Fig. 12 shows the predicted extrudate temperature from this


initial billet temperature profile and the previously developedram speed trajectory. Finally, the required initial axial billettemperature is shown in Fig. 13.

VII. CONCLUSION

We have presented a model which is derived directlyfrom the mathematical description of the physical phenomenapresent. This model has demonstrated real-time performance.The construction of the model required only a minimum ofassumptions.

The resulting temperature state equation is bilinear in thetemperature state and the control, ram velocity plus nonlinearin the billet length state (measurable).

This paper successfully demonstrates the use of a physicallymotivated model for identification and open-loop control of theextrusion process. For this example, we picked one particularobjective function for the open-loop design. The model formcan support many different objective functions or constraintconditions; therefore, the model can be useful for a widevariety of products.

Further refinements are required; particularly noteworthy isthe change in the flow stress model. Other future developmentsmay include investigation of excitation persistency for identi-fication plus observer and feedback controller development.

ACKNOWLEDGMENT

The authors would like to thank the Werner Co. for per-forming the identification plant trials and Williamson Corp.providing a temperature sensor for these trials. They wouldalso like to thank M. Lefstad, SINTEF, University of Trond-heim, for allowing the use of his extrusion data.

REFERENCES

[1] B. Tibbetts and J. Wen, “Application of modern control and modelingtechniques to extrusion processes,” inProc. 4th IEEE Conf. on Contr.Applicat., 1995, pp. 85–90.

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[13] K. Mori, “Extrusion and drawing,” inNumerical Modeling of MaterialDeformation Processes, P. Hartley, I. Pillinger, and C. Sturgess, Eds.London: Springer-Verlag, 1992, pp. 303–317.

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Brian R. Tibbetts (M’97) received the B.S. de-gree in mechanical engineering from RensselaerPolytechnic Institute, Troy, NY, in 1986 and theM.S. degree in electric engineering at the Universityof Texas at Arlington in 1993. He is currentlypursuing the Ph.D. degree in computer and systemsengineering at Renselaer on a U.S. Dept. of EnergyFellowship for Integrated Manufacturing for workon aluminum extrusion process control.

He worked for General Dynamics, Fort WorthDivision/Lockheed Fort Worth from May 1986 to

January 1994. During this period, he worked on many aspects of flightcontrol hardware and system architecture on a variety of concept and prototypeaircraft, notably the YF-22. He returned to Rensselaer in January 1994 andjoined the N.Y.S. Center for Advanced Technology in Automation, Robotics,and Manufacturing. His current research interests include modeling, control,and software architecture for systems and processes.

John Ting-Yung Wen (S’79–M’81–SM’93)received the B.Eng. degree from McGill University,Montreal, Canada, in 1979, the M.S. degree fromthe University of Illinois, Urbana-Champaign,in 1981, and the Ph.D. degree from RensselaerPolytechnic Institute, Troy, NY, in 1985, all inelectrical engineering.

From 1981 to 1983, he was a System Engineerwith Fisher Control Company, Marshalltown, IA,where he worked on the coordination control of apulp and paper plant. From 1985 to 1988, he was

a member of technical staff at the Jet Propulsion Laboratory, Pasadena, CA,where he worked on the modeling and control of large flexible structures andmultiple-robot coordination. Since 1988, he has been with the Departmentof Electrical, Computer, and Systems Engineering at Rensselaer PolytechnicInstitute, where he is currently a Professor. He is also a member of theNew York Center for Advanced Technology in Automation, Robotics andManufacturing. His current research interests are in the area of modelingand control of multibody mechanical systems, including flexible structures,robots, and vehicles, and material processing systems such as extrusion andresistive welding.

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