extrusion march98

8/6/2019 Extrusion March98

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

Extrusion Process Control: Modeling,Identication, 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 deformationspecically, extru-sion. A process model is required for control. We state the difcul-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 identication 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 theidentication and open-loop optimization problems are tractable.An example with plant trial data is provided.

Index Terms Approximation methods, identication, 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 the dummy block (2), transmittingthe force to the billet (5). Initially, this force causes upsetthewhole billet is deformed to t the container (3) shape andthere is initial ow from the die (4). After upset, the extrusionprocess takes place, producing the desired extrudate (6). Themetal for the extrudate comes from the deformation zone(5a) within the billet. The material within the dead metal zone (5b) remains within the container throughout the process.During the entire process, the material ow and shape of thedeformation and dead metal zones are evolving. After most of the billet has been extruded, the ow 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 York State 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 Identier S 1063-6536(98)01819-3.

Fig. 1. Schematic of direct extrusion.

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

stock shapes for secondary processingproduction ef-ciency is very important since these items have a com-modity pricing structure;

furniture componentssurface quality is key; architectural componentssurface 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 a

wide 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 identier 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 wheresimplications are appropriate which are not broadly applica-

10636536/98$10.00 1998 IEEE


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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 equation

for heat transfer. The governing equation is

(4)

with the following boundary conditions:

where is the thermal diffusivity, is the density of aluminum, is the specic 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 applynite-difference or spectral approximation to obtain a solutionaccording to the following rules:

The governing equation becomes

(5)

where the ow velocity is represented as

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

wherezeroth order Bessel function, rst kind;rst-order Bessel function, rst kind;zeros of .

Inner Product:

Eigenvalue Problem #2:

Solution:

where Inner Product:

The original equation, (5), is transformed from a partialdifferential equation to a nite system of ordinary differentialequations. Equation (5) is projected onto the basis set of thetemperature approximation using the inner product operationsdened previously. Integration by parts is used to exchangehigher order derivatives of the temperature eld for higherorder derivatives of the basis vectors. The stabilizing inuenceof the conduction term permits the innite dimensional systemof ODEs 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 nal state equations become

(7)


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TIBBETTS AND WEN: EXTRUSION PROCESS CONTROL 137

where

. . .

... .... . .

is the projected temperature state, is coefcientmatrix for the conduction term, is the coefcient matrixfor the advective term (rate of change in temperature due tomotion of a bulk with a temperature gradient), is thecoefcient 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 the

ram position (measurable). Temperature evolution is bilinear with respect to the

temperature state, , and the control, . There are free coefcients in and which

may be determined by through identication. The elements of and are determined solely

by 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 innite-dimension system of equations tobe truncated to a very low order.

B. Boundary Condition Denition

The model given in Section III-A is for the generalizedboundary condition, . In this section, themodel is made more specic by chosing the gradients tobe independent of temperature. Three gradient functions arespecied 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 thatspecic parameters are available for identication 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 conguration 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 coefcients represent the magnitude of thesefunctions.

C. Heat Generation Term Denition

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 ow stress. Therefore, the heat generation term can


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be represented as

(10)

The specic 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 coefcients, , 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 or

an 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 of the 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 coefcients 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 specic load functions, , , and , will bedeveloped in Section IV. A bias term has 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 dened 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 ow eld 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)

whereprojected 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.


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IV. F LOW FIELD DEVELOPMENT

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

We chose upper bound analysis (UBA) as the basis of thedeformation model component. UBA is the search for thekinematically admissible ow eld which requires the

minimum powers.t.

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

Han et al. [20] developed a ow eld approximation foruse in UBA for a contoured die, very light reduction extrusionproblem. This particular ow eld did appear to be a closerapproximation to the physical situation than the typical UBAow eld. The ow eld includes a bulk component (xed)and a variational component. The parameters of the variationalcomponent of the ow eld 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 ow boundary forapplication to at-faced dies, , shown in Fig. 3.

Generalized the structure of the variational component of the ow eld.

Determined required conditions on the ow eld 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 ow eld descrip-tion. The algorithm includes line searches, modied linesearches, and steepest descent methods. Increases in speedand robustness against local minima are observed.

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

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

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

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

(14)

subject to the following constraints:

where

The strain rate tensor components are dened as

and the strain rate magnitude, , is dened as

The power optimization is described by (15) where isthe minimum power resulting from a ow eld 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 ow eld and, therefore, on the


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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), and is the materialow stress.

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

linearly

From (17), the load terms are dened as

The velocity eld, via the denitions for the extrusion load,is used to construct the heat generation terms which result

from 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 dened throughout the volume. The introductionof the Dirac delta function and application of appropriatescaling permit these function to be used in the spectralapproximation.

V. I DENTIFICATION

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

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

depends on 1) the ow eld (xed by constructionof the deformation model component) and 2) the material owstress and coefcients of friction and shear. The coefcients of friction and shear are not specied and, therefore, suitable foridentication. The ow stress is also suitable for identicationsince it appears as a constant. This constant represents anaggregate of the constitutive relationship operating on the oweld and the temperature distribution.

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

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

(17)

The parameters for identication are expressed for conve-nience as

A. Estimated Extrudate Temperature

Identication will use measured temperature, , and itsestimate, . This measurement is discrete with samples.Therefore, and

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

(18)

where is the output matrix,and are the number of


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basis functions in and , respectively, and

...

......

where

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

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

be identied, . This linearity will be maintained as long as: ow stress is not treated as an explicit function of

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

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 ow eld for the extrusion geometry andtabulated by current billet length (off-line).

2) Calculate and which depend on the oweld (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 (8 8 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 Identication

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


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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. Theidentication problem requires that the following minimizationbe solved [22]:

(20)

where is the set of permissible parameter values and the

cost function is dened 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 reective relative condence 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 identication and the result modelpredictions are shown in Fig. 6. The comparison of actual dataand model prediction with errors generally below 10%. A trend

is noted in the temperature prediction being positively biasedand the load being negatively biased.

The identied parameters were used in the model with a newplant trial data set to gage the success of the identicationprocedure. The extrusion which generated this new data setused a different initial billet temperature prole 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 identication run. The error in load predictionhas decreased.


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

Fig. 7. Model predictions for the reference extrusion.

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

Both the identication run and the verication 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 ow 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 70 Chotter than the billet used for identication. Also, the extrudatetemperature was 20 C hotter. Therefore, the ow stress of thisbillet will substantially lower than the identication billet, thusproducing less heat during extrusion. To see the effects, theverication run was rerun with a 15% reduction in the owstress parameter. The results are shown in Fig. 10. We see that

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

Fig. 9. Comparison between identication and reference extrusions: Initialbillet temperature prole.

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

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

One source of difculty for using this data set for identica-

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 identication, 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 specic values of the physical processvariables [9], [18], [5]. We have chosen constant maximum


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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 nd an initial billet temperatureprole 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 nished approximately 10% faster thanthe extrusion shown in the previous section for identicationvalidation. The ram speed proles 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 temperatureproles.

TABLE IIIOPTIMIZATION: INITIAL BILLET TEMPERATURE MODE CONSTRAINTS AND VALUES

To determine the desired initial billet temperature prole,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 identication problem fordetermining is motivated since enters into (23) linearly.

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

(24)

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

(25)

Exactly as was done for the identication 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 prolecould be imposed axially and that there is no radial variability.The rst 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 specied at 520 C. The requiredinitial projected temperature modes are given in Table III.Fig. 12 shows the predicted extrudate temperature from this


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initial billet temperature prole 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 of assumptions.

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 identication 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 renements are required; particularly noteworthy isthe change in the ow stress model. Other future developmentsmay include investigation of excitation persistency for identi-cation plus observer and feedback controller development.

ACKNOWLEDGMENT

The authors would like to thank the Werner Co. for per-forming the identication 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.

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[18] T. Sheppard, The application of limit diagrams to extrusion processcontrol, in Proc. 2nd Int. Alum. Extrusion Tech. Sem. , 1977, vol. 1, pp.331337.

[19] W. Prager and P. G. Hodge, Theory of Perfectly Plastic Solids . NewYork: Wiley, 1951.

[20] C. H. Han, D. Y. Yang, and M. Kiuchi, A new formulation forthree-dimensional extrusion and its application to extrusion of cloversections, Int. J. Mech. Sci. , vol. 28, no. 4, pp. 201218, 1986.

[21] M. Lefstad, Metallurgical speed limitations during the extrusion of AlMgSi-alloys, Ph.D. dissertation, Univ. Trondheim, Norway, Nov.1993.

[22] L. Ljung, System Identication:Theory for the User . Englewood Cliffs,NJ: Prentice-Hall, 1987.

Brian R. Tibbetts (M97) 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 work

on 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 ightcontrol hardware and system architecture on a variety of concept and prototypeaircraft, notably the YF-22. He returned to Rensselaer in January 1994 and joined 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 (S79M81SM93)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 exible 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 exible structures,robots, and vehicles, and material processing systems such as extrusion andresistive welding.

extrusion march98

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