Extrusion Process Control - Semantic Scholar

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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 methodology for process control of bulk deformation—specifically, extrusion. A process model is required for control. We state the difficulties which exist with currently available models. An appropriate decomposition and associated assumptions are introduced which permit a real-time process model to be constructed via spectral approximation to solve the axi-symmetric two-dimensional (2D) transient heat conduction equation with heat generation and loss. This solution, simulation results, and model execution times are provided. We use this model development plus output relationships from previous model development work to develop parameter identification and open-loop control methodologies. These methodologies are motivated by the structure and properties of the developed model. We demonstrate that the parameters and control variables enter into the model equations so that the identification and open-loop optimization problems are tractable. An example with plant trial data is provided. Index Terms— Approximation methods, identification, metal industry, modeling, optimization methods, partial differential equations, process control, spectral approximation.

I. THE PROCESS

T

HE direct extrusion process is illustrated in Fig. 1. The ram (1) applies force to the dummy block (2), transmitting the force to the billet (5). Initially, this force causes upset—the whole billet is deformed to fit the container (3) shape and there is initial flow from the die (4). After upset, the extrusion process takes place, producing the desired extrudate (6). The metal 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 flow and shape of the deformation and dead metal zones are evolving. After most of the billet has been extruded, the flow contorts to such a degree that extrusion is no longer feasible. The remainder of the billet is 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 was supported by the Aluminum Processing Program at Rensselaer Polytechnic Institute, a consortium of ten Aluminum Extruders, a U.S. Department of Energy Fellowship for Integrated Manufacturing, and the New York State Center for Advanced Technology (CAT) in Automation, Robotics, and Manufacturing. The CAT is partially funded by a block grant from the New York State Science and Technology Foundation. B. R. Tibbetts is with the New York State Center for Advanced Technology in Automation, Robotics, and Manufacturing, Rensselaer Polytechnic Institute, Troy, NY 12180-3590 USA. J. T.-Y. Wen is with the Electrical, Computer, and Systems Engineering Department, 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. Some specific examples are: • stock shapes for secondary processing—production efficiency is very important since these items have a commodity pricing structure; • furniture components—surface quality is key; • architectural components—surface quality and microstructural properties are both critical since strength and aesthetics are required. The goal of our work is to develop cost-effective strategies for control of this process. These strategies need to address a wide range of product geometries, processing conditions, qualities requirements, and productivity goals. A process model is required to achieve this goal. The process model enters explicitly into an identifier or observer, operating in real-time plus it may be directly incorporated into a feedback controller. Therefore, this model must be designed to capture the response of very different systems (for example, round rod with area reduction of ten times or rectangular tubing with area reduction of 500 times) and require modest computational power. II. BACKGROUND Process control of bulk deformation has received relatively little attention from the control systems community. Much of the work found has been limited by the existing models. For example, Meyer and Wadley [3] developed a modelbased feedback control for material properties and inequality constraints for part yielding. This approach could only be applied to relatively slow processes due to model complexity; in this particular case, the hot isostatic process required 2 h to perform. Previously reported work has focused on problems where simplifications are appropriate which are not broadly applica-

1063–6536/98$10.00  1998 IEEE

TIBBETTS AND WEN: EXTRUSION PROCESS CONTROL

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ble. For example, there has been an ongoing effort in control of isothermal thermal forging. The researchers involved include J. 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 of work [4]–[10] assumes that there is no temperature change in the workpiece during the process. The condition does not apply to metal extrusion and is not expected to be applicable to many other deformation processes such as drawing. Also, it is notable that even with the above the simplifications, the resulting models were not suitable for real-time control. There has been some work which is based on removing the need for a model. The previously cited group proposed using artificial neural networks to emulation the complex finite element method (FEM)-based model and open-loop optimization [7], [9], [10]. This effort did not remove the constant temperature restriction or faced the prospect of computationally expensive network retraining. Another example is cyclic learning control for extrusion developed by Pandit and Buchheit [11], [12]. This work employed a black-box model and required strong assumptions on process linearity, actuation behavior, and constancy of disturbances. Many typical deformation processes will violate one or more of these assumptions. Based on our above assessment of the available work and its possible application to extrusion, an existing model was sought. These models could be classified into two broad areas: FEM-based models (detailed) or averaged-variable models (coarse, approximation-based). The FEM-based models provide the needed information, but cannot be implemented in real time [13]–[15]. The averaged-variable models are computationally tractable; however, they do not provide localized information needed for control [16], [17]. Therefore, we chose to develop a model of the extrusion process specifically for real-time control.

III. THE MODEL

FOR

CONTROL DESIGN

The thermo-mechanical response of the billet during extrusion is governed by a set of nonlinear, coupled partial differential equations given by 1) heat transfer and 2) deformation approximation (1) s.t.

(2)

These equations are coupled via the flow stress of the material which is part of the heat generation expression and . the deformation power Our purpose requires that these equations be solved; however, direct solution cannot be done in real time. The following assumptions are introduced. 1) The deformation field determined with constant flow stress will closely approximate the optimum field in the set of possible fields.

Fig. 2. Thermo-mechanical model components.

2) The set of possible fields can be restricted to include only fields which are linear in ram velocity. 3) Model errors introduced by using nonlocal flow stress are acceptable. Assumption 1) accomplishes the decoupling of (1) and (2) by not considering the temperature dependence of the material flow stress. Assumption 2) allows the model description to be independent 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 is shown in Fig. 2. Development of the heat transfer and deformation components are described below. The heat transfer model component establishes the extrusion model framework or basic model structure. The deformation model component, in conjunction with the structure of the heat transfer model, provides the unique attributes for a given extrusion. The material response can be included directly by using the temperature predicted by the heat transfer model and the strain rate predicted by the deformation model to determine an appropriate flow stress value. This flow stress value, in turn, influence the subsequent temperature distributions. Our model is based on an axi-symmetric extrusion, round billet to round extrudate. This restriction has been chosen since even with the simplifications introduced thus far, there is little hope of a real-time solution for an arbitrary shape. We expect that the analytical model will capture a large portion of the response for an arbitrary extrusion shape and the black-box model will capture the remainder. This problem is naturally described in cylindrical coordinates. The -axis is the centerline through the center of the die port and the billet. The -axis extends radially from the -axis. The origin is at the die entrance. The current billet length (positive direction) is denoted by . The coordinate system, main geometric parameters, and the flow boundary function 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 flow velocity field from the ram velocity; however, no specific form will be assumed. The next section will review the basis for (2) and the development of one possible family of flow velocities which satisfy this equation. A. State Equation Development There are two physical quantities from which the states and their evolution equations are developed: the billet size and the billet temperature distribution. The system input is the ram

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in ) allows the desired boundary conditions to be imposed. The temperature is approximated by (6) based on the following eigenvalue problems. Eigenvalue Problem #1:

Fig. 3. Problem coordinates.

Solution:

speed, . The billet size state can be represented by the billet length, , with the evolution equation where (3) Development of the states and evolution equation for the billet temperature distribution begins with the governing equation for heat transfer. The governing equation is

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

(4) with the following boundary conditions:

Eigenvalue Problem #2:

where is the thermal diffusivity, is the density of aluminum, is the specific heat capacity, is temperature is the heat generation term, and is represents the billet boundary. The changing boundary and the loss of mass via the extrudate adds an interesting component to this problem. We have transformed this equation so that one can apply finite-difference or spectral approximation to obtain a solution according to the following rules:

Solution:

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 infinitedimension system of ordinary differential equations. An appropriate set of basis functions (Bessel functions in and cosines

where Inner Product:

The original equation, (5), is transformed from a partial differential equation to a finite system of ordinary differential equations. Equation (5) is projected onto the basis set of the temperature approximation using the inner product operations defined previously. Integration by parts is used to exchange higher order derivatives of the temperature field for higher order derivatives of the basis vectors. The stabilizing influence of the conduction term permits the infinite dimensional system of ODE’s to be truncated. Integration by parts of the conduction term also requires that the temperature gradient boundary conditions be incorporated into the solution. Gradient boundary conditions which consist of temperature-dependent and constant terms can be incorporated. After the above manipulation and rearrangement into a matrix form, the final state equations become

(7)


137

dummy block

where

The heat transfer boundary term is derived from three , , and , which characterize the functions, boundary. The following decomposition is introduced so that specific parameters are available for identification and the parameters enter into the equation linearly

Using the above decomposition, the boundary condition in the temperature state equation can be exterm pressed as .. .. .

.

(8)

.. . ..

.

is the “projected temperature state”, is coefficient matrix for the conduction term, is the coefficient matrix for the advective term (rate of change in temperature due to motion of a bulk with a temperature gradient), is the coefficient matrix for the heat generation term, is the boundary 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 coefficients in and which may be determined by through identification. • The elements of and are determined solely by problem geometry and the inner product operation for the spectral approximation. • The elements of and are determined by the deformation component , and and the spectral approximation. • The stabilizing, conduction term dominates as higher order basis functions are included in the approximation, permitting the infinite-dimension system of equations to be truncated to a very low order.

where

The variable represents the configuration adjustable component of the boundary conditions. For example, the simulation results shown subsequently assume perfect insulating conditions at the extrudate and constant leaky conditions at the die. The coefficients represent the magnitude of these functions. C. Heat Generation Term Definition A similar decomposition can be performed on the heat generation term . This term is constructed from deformation, friction, and shearing sources; the following equation illustrates this calculation:

(9) The heat generation functions have the following dependencies:

B. Boundary Condition Definition The model given in Section III-A is for the generalized boundary condition, . In this section, the model is made more specific by chosing the gradients to be independent of temperature. Three gradient functions are specified at 1) the container wall; 2) the die face; and 3) the

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

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represented as follows:

be represented as

(10) (11) The specific heat generation functions, , will be developed in Section IV.

,

, and

D. Output Equations There are many possible output relations which one can derive from this model. For our purposes, we will present three particular equations, the extrudate temperature, the extrusion load, and the maximum strain rate within the billet. 1) Extrudate Temperature: The output equation for the extrudate temperature depends greatly on the method of measurement or on the use of the variable. Generically, this variable can be calculated as follows:

If one wishes to consider the process performance or a research setup which uses an instrumented die with a thermal couple, then the coefficients, , are time-invariant and can be written as . The typical measurement capability in the production environment requires a more complicated 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 devices will measure temperature on the extrudate where the extrudate exits the die bolster (a substantial support structure which holds the die in place). This point is approximately 3 to 4 ft from the die exit. A very thorough description would require that one consider the continued conduction throughout the extrudate, 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 coefficients must contain a time delay, dependent on , and the projection of the billet modes at the die face onto the zeroth radial mode of the extrudate

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

The specific load functions, , , and , will be developed in Section IV. A bias term has also been included to account for various losses such as friction in the press and errors in load measurement. 3) Maximum Strain Rate: Our investigation into the phenomenological effects of the extrusion process on the quality of 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 this local strain rate should be defined so that one has a parameter which is a good indicator of product quality. We, therefore, suggest that the maximum strain rate within the billet be used as an approximation to this indicator. By construction of the flow field description (Section IV), this parameter is linear in (12) where rate.

represents the functional description of the strain

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 to motion 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 ram position and billet location, to strain rate.


IV. FLOW FIELD DEVELOPMENT We have fully developed the deformation model component in earlier work [1], [2]. A brief summary of these results is provided for completeness. We chose upper bound analysis (UBA) as the basis of the deformation model component. UBA is the search for the kinematically admissible flow field which requires the minimum power

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functions without affecting the bulk material flow. This result is given below

s.t. According to the UBA theorem [19], this flow field is the actual flow field. Conventional applications of UBA were specifically for determining required process power. Typically, simple flow fields have been used which predict the power; however, they bare little resemblance to the physical flow fields observed experimentally. This is not suitable for our purpose. Han et al. [20] developed a flow field approximation for use in UBA for a contoured die, very light reduction extrusion problem. This particular flow field did appear to be a closer approximation to the physical situation than the typical UBA flow field. The flow field includes a bulk component (fixed) and a variational component. The parameters of the variational component of the flow field are chosen to minimize the process power via a numerical optimization algorithm. We have made the following improvements in the original work for application to typical industrial extrusion control problems. • Determined required conditions on the flow boundary for application to flat-faced dies, , shown in Fig. 3. • Generalized the structure of the variational component of the flow field. • Determined required conditions on the flow field description so that it is linear in ram velocity (the control variable). • Implemented a hybrid optimization algorithm for determining the free parameters in the flow field description. The algorithm includes line searches, modified line searches, and steepest descent methods. Increases in speed and robustness against local minima are observed. With the conditions specified, the optimization can be performed off-line, independent of the control. The conventional UBA applied to flat-face dies required that the material ultimately going into the die (region 5a in Fig. 1) be artificially divided into an undeformed region (far from the die) and a deformation zone. This division is difficult to specify and generates local discontinuities. These difficulties are avoided with our approach. We demonstrated good qualitative prediction of experimental measurements of a flow field with our deformation model. The kinematically admissible velocity field description is constructed by forming the axial component, , first and then directly calculating the radial velocity component . The first term (bulk component) in the axial velocity function guarantees that the same amount material flows through all axial cross sections. The remaining terms with constraints are constructed to locally modify the flow via polynomial basic

(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 is the minimum power resulting from a flow field described in the previous section with the most favorable set of free parameters and boundary description, (15) There are three contributing factors to the power used during extrusion: 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 terms depend on the velocity of the flow field and, therefore, on the

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free parameters introduced above. The power components are

(16) is the height of the dead zone, , , and where are the same scaling factors as in (10), and is the material flow 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

which are not specifically defined by the modeling effort. The conduction and advective terms in the temperature state equation (13a) are completely defined by the modeling process; the conduction term is solely a function of the billet geometry and the advective term is defined by the flow field. The heat boundary condition describes the gradient at the boundary and is not specified. The heat generation term depends on 1) the flow field (fixed by construction of the deformation model component) and 2) the material flow stress and coefficients of friction and shear. The coefficients of friction and shear are not specified and, therefore, suitable for identification. The flow stress is also suitable for identification since it appears as a constant. This constant represents an aggregate of the constitutive relationship operating on the flow field and the temperature distribution. The ram position state transition equation (13b) does not have parameters for identification. Similarly, the temperature output equation (13c) does not have parameters requiring identification since this equation depends only on the projected temperature state and the workpiece geometry. The extrusion load equation does have the same parameters for identification that the heat generation term has. The load equation can be defined in terms of the heat generation term

(17)

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

The parameters for identification are expressed for convenience as

A. Estimated Extrudate Temperature Identification will use measured temperature, . This measurement is discrete with estimate, Therefore,

The heat generation kernel due to deformation is the integrand from the extrusion load analog. The heat generation kernels due to friction and shear are slightly more complicated since they are not defined throughout the volume. The introduction of the Dirac delta function and application of appropriate scaling permit these function to be used in the spectral approximation.

, and its samples. and

The estimate for measured temperature is calculated by using the forth-order Runge–Kutta integration algorithm to transform the continuous state equation (13a) into a discrete representation (18)

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

where

is the output matrix, and are the number of


basis functions in

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and , respectively, and

.. .

.. .

.. .

where

Fig. 4. Predicted extrudate temperature.

Also, note that represents the conduction matrix at the th instantance . Similarly, represents the advection matrix at the th 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 of temperature (an assumption in the model development); • boundary temperature gradients are not temperature dependent. The linearity of this relationship greatly improves the solvability of the problem over a typical nonlinear description. Equation (18) is linear in the initial state, , also. This property will prove very useful in solving the open-loop control 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 the model, including predicted internal billet temperature distribution. Next, our initial parameter identification technique for this model is shown. Finally, using the equations developed for the identification, we solve an open-loop control problem. Experimental data from an experimental extrusion press and from a production press is used for comparison and identification. A. Model The temperature predictions shown in Fig. 4 and in Fig. 5 are based on experimental extrusion of AA6060 by Lefstad [21]. The billet is 16.5-cm length and 10.0-cm diameter. The extrudate is 1.04-cm diameter and the ram velocity throughout the push is 2.0 cm/s. Experimentally measured extrudate temperature is shown in Fig. 4.

The model-generated results were produced in the following manner. 1) Constructed a flow field for the extrusion geometry and tabulated by current billet length (off-line). 2) Calculate and which depend on the flow field (off-line). and , (off-line). 3) Calculate 4) Solve the temperature evolution equation (on-line). These results reasonably approximate the experimental measurements for most of the extrusion. Also, the model predicts that the core will be cooler than the extrudate surface, consistent with available information. The isothermal plots for different times during the push show that the majority of the temperature increase is generated by the intense shearing between the deformation zone and dead zone, regions 5a and 5b in Fig. 1, respectively. Note, the heavy boundary represents the current billet shape and the extrudate size (toward the bottom). The off-line calculations for the simulation shown required 2 h with a 586, 100 MHz running Windows NT 3.51. The results 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 are present in a 16-state model (4 4 basis functions). The reduced-state model for this example executes in 3.0 s on the same platform. Although some care was exercised in code development to optimize for speed, much more can be done for a real-time implementation. Based on number of operations and raw processor performance, improvements up to a factor of ten might be realizable on this platform. In addition, lowcost hardware alternatives exist which might realize similar improvements. Therefore, the high-resolution model can be implemented in real-time. B. Parameter Identification The identification problem will use both measured temperature 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 II PARAMETER IDENTIFICATION: PARAMETER VALUES

TABLE I PARAMETER IDENTIFICATION: PARAMETER CONSTRAINTS

a constrained quadratic optimization problem. Earlier efforts used a least squares approach which did generate parameters; however, these parameters were not physically realizable. The identification problem requires that the following minimization be solved [22]: (20) is the set of permissible parameter values and the where cost function is defined as (21) where represents the weighting on the measured temperature error and is the weighting on the extrusion load error. The weighting parameters have two purposes. First, they can be used to reflective relative confidence in the measurement accuracy. Second, the force and temperature measurement have dissimilar units and different magnitudes. These differences imply a weighting factor, perhaps an inappropriate weighting. Explicit inclusion of a weighting factor and an expectation of

the magnitudes of the different data items permits adjustment of the weighting parameters for a balanced cost function. The nonlinear programming algorithm, constr() in MATLAB was used with the constraints shown in Table I to minimize the cost function. The weightings in the cost function were set with temperature measured in degrees Celius and and . force in kiloNewtons: The resulting parameter values are provided in Table II. The plant data used for the identification and the result model predictions are shown in Fig. 6. The comparison of actual data and model prediction with errors generally below 10%. A trend is noted in the temperature prediction being positively biased and the load being negatively biased. The identified parameters were used in the model with a new plant trial data set to gage the success of the identification procedure. The extrusion which generated this new data set used a different initial billet temperature profile and a different ram speed trajectory. Fig. 7 shows the experimental data and the model predictions. The prediction of trends and general agreement are good. Although still within 10%, the error in temperature prediction has increased compared with the error present in the identification run. The error in load prediction has decreased.


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Fig. 8. Comparison between identification and reference extrusions: Ram speed trajectory.

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

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

Fig. 7. Model predictions for the reference extrusion.

Figs. 8 and 9 illustrate the differences in ram speed trajectory and initial billet temperature profile between the extrusion used for identification and the extrusion used as reference to check the identification. Both the identification run and the verification run show temperature predictions above the measured temperature. We believe that there are two sources of this discrepancy. First, improvements in the deformation model for short billets are require. In general, the force predictions for short billets are too high which will result in elevated temperature prediction. Second, constant flow stress is not accurate and is expected to have a substantial impact on the temperature prediction. The validation run started with a billet which was 40 to 70 C hotter than the billet used for identification. Also, the extrudate temperature was 20 C hotter. Therefore, the flow stress of this billet will substantially lower than the identification billet, thus producing less heat during extrusion. To see the effects, the verification run was rerun with a 15% reduction in the flow stress parameter. The results are shown in Fig. 10. We see that

Fig. 10. Model predictions for the reference extrusion with reduced flow stress.

this validation run has significantly less error than the original validation run. One source of difficulty for using this data set for identification is the lack of variation in the input signal (the ram speed, shown in Fig. 8). It is highly likely that this signal does not satisfy the persistency of excitation condition, resulting in only partial identification, rather than unique determination, of the parameters. C. Open-Loop Control The range of extruded products and their applications are large. As such, the applicable measures of quality and their required levels vary from product to product. These requirements can be mapped to specific values of the physical process variables [9], [18], [5]. We have chosen constant maximum

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Fig. 13. Comparison of the optimized and reference initial billet temperature profiles. TABLE III OPTIMIZATION: INITIAL BILLET TEMPERATURE MODE CONSTRAINTS

AND

VALUES

Fig. 11. Comparison of the optimized and reference ram speed and maximum strain rate trajectories.

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

Fig. 12. Predicted extrudate temperature using optimized ram speed trajectory and initial billet temperature. The reference extrudate temperature is also shown.

strain rate and constant extrudate temperature to represent good quality for this example. The maximum strain rate, given by (13e), is linear in the ram speed, ; however, it is nonlinear in the billet length. Also, it is not dependent on the projected temperature state variables. The temperature evolution equation (13a) depends on both the initial projected temperature state and the ram speed . Therefore, the suggested design procedure is trajectory to determine the desired ram speed trajectory which results in the appropriate maximum strain rate. With the ram speed determined, one can then find an initial billet temperature profile which achieve a constant extrudate temperature for the ram speed trajectory. The required ram speed trajectory can be calculated easily since 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 our optimized extrusion finished approximately 10% faster than the extrusion shown in the previous section for identification validation. The ram speed profiles and the associated maximum strain rate predictions for the referenced plant trial and the open-loop control design are shown in Fig. 11.

(23) Note, we have expressed the implicit dependence of the and on the ram speed trajectory. Since ram speed is known, all parts of (23) are known except for . An approach analogous to the parameter identification problem for determining is motivated since enters into (23) linearly. Finding the best initial billet temperature profile requires that the following minimization be solved: (24) where is the set of permissible initial projected temperature states and the cost function is defined as (25) Exactly as was done for the identification problem, the nonlinear programming algorithm, constr() in MATLAB was used with the constraints shown in Table III to minimize the cost function, . The constraints on the initial billet temperature were driven by the actuation capabilities. We assumed that the best one could hope is that a “reasonably smooth” profile could be imposed axially and that there is no radial variability. The first six states represent the constant radial variable axial basis functions. Since the greater index represents a mode with higher spatial frequency, these modes were subject to tighter restrictions to impose the “reasonably smooth” criteria. The control problem was solved with desired extrudate temperature being arbitrarily specified at 520 C. The required initial projected temperature modes are given in Table III. Fig. 12 shows the predicted extrudate temperature from this


initial billet temperature profile and the previously developed ram speed trajectory. Finally, the required initial axial billet temperature is shown in Fig. 13. VII. CONCLUSION We have presented a model which is derived directly from the mathematical description of the physical phenomena present. 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 the temperature state and the control, ram velocity plus nonlinear in the billet length state (measurable). This paper successfully demonstrates the use of a physically motivated model for identification and open-loop control of the extrusion process. For this example, we picked one particular objective function for the open-loop design. The model form can support many different objective functions or constraint conditions; therefore, the model can be useful for a wide variety of products. Further refinements are required; particularly noteworthy is the change in the flow stress model. Other future developments may include investigation of excitation persistency for identification plus observer and feedback controller development. ACKNOWLEDGMENT The authors would like to thank the Werner Co. for performing the identification plant trials and Williamson Corp. providing a temperature sensor for these trials. They would also like to thank M. Lefstad, SINTEF, University of Trondheim, for allowing the use of his extrusion data. REFERENCES [1] B. Tibbetts and J. Wen, “Application of modern control and modeling techniques to extrusion processes,” in Proc. 4th IEEE Conf. on Contr. Applicat., 1995, pp. 85–90. , “Control framework and deformation modeling of extrusion [2] processes: An upper bound approach,” in Proc. 6th Int. Alum. Extrusion Tech. Sem., 1996, vol. 1, pp. 375–386. [3] D. Meyer and H. Wadley, “Model-based control feedback control of deformation processing with microstructure goals,” Metallurgical Trans. B, vol. 24B, pp. 289–300, Apr. 1993. [4] A. Kumar, R. Grandhi, A. Chaudhary, and D. Irwin, “Processing modeling and control in metal forming,” in 1992 Amer. Contr. Conf., Chicago, IL, June 1992, pp. 817–821. [5] J. Malas III and V. Seetharaman, “Using material behavior models to develop process control strategies,” JOM, vol. 44, pp. 8–13, June 1992. [6] R. V. Grandhi, A. Kumar, A. Chaudhary, and J. Malas III, “State-space representation and optimal control of nonlinear material deformation using the finite element method,” Int. J. Numerical Methods, vol. 36, pp. 1967–1986, June 1993. [7] J. Berg, J. Adams, J. Malas III, and S. Banda, “Design of ram velocity profiles for isothermal forging via nonlinear optimization,” in 1994 Amer. Contr. Conf., Baltimore, MD, June 1994, pp. 323–327. [8] J. Berg, J. Malas III, and A. Chaudhary, “Open-loop control of a hotforming process,” in Proc. 5th Int. Conf. on Numerical Methods in Industrial Forming Processes—NUMIFORM’95, Ithaca, NY, June 1995, pp. 539–544. [9] J. Berg, R. Adams, J. Malas III, and S. Banda, “Nonlinear optimizationbased design of ram velocity profiles for isothermal forging,” IEEE Trans. Contr. Syst. Technol., vol. 3, pp. 269–278, Sept. 1995. [10] C. A. Schwartz, J. M. Berg, R. Chang, and M. Mears, “Neuralnetwork control of an extrusion,” in 1995 Amer. Contr. Conf., 1995, pp. 1782–1786.

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[11] M. Pandit and K. Buchhei, “Control schemes for a cyclically operating aluminum extruder plant,” in Proc. 3th IEEE Conf. on Contr. Applicat., 1994. [12] M. Pandit and K. Buchheit, “A new measurement and control system for isothermal extrusion,” in Proc. 6th Int. Alum. Extrusion Tech. Sem., 1996, vol. 1, pp. 79–86. [13] K. Mori, “Extrusion and drawing,” in Numerical Modeling of Material Deformation Processes, P. Hartley, I. Pillinger, and C. Sturgess, Eds. London: Springer-Verlag, 1992, pp. 303–317. [14] E. Atzema and J. Huetink, “Finite element analysis of forward–backward extrusion using ALE techniques,” in Proc. 5th Int. Conf. on Numerical Methods in Industrial Forming Processes—NUMIFORM’95, Ithaca, NY, June 1995, pp. 383–388. [15] D. Vucina and I. Duplancic, “Flow formulation FE metal forming analysis with boundary friction via penalty function,” in Proc. 6th Int. Alum. Extrusion Tech. Sem., 1996, vol. 1, pp. 69–72. [16] K. Laue and H. Stenger, Extrusion. Metals Park, OH: Amer. Soc. Metals, 1976. [17] G. Dieter, Mechanical Metallurgy, 3rd ed. New York: McGraw-Hill, 1986. [18] T. Sheppard, “The application of limit diagrams to extrusion process control,” in Proc. 2nd Int. Alum. Extrusion Tech. Sem., 1977, vol. 1, pp. 331–337. [19] W. Prager and P. G. Hodge, Theory of Perfectly Plastic Solids. New York: Wiley, 1951. [20] C. H. Han, D. Y. Yang, and M. Kiuchi, “A new formulation for three-dimensional extrusion and its application to extrusion of clover sections,” Int. J. Mech. Sci., vol. 28, no. 4, pp. 201–218, 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 Identification: Theory for the User. Englewood Cliffs, NJ: Prentice-Hall, 1987.

Brian R. Tibbetts (M’97) received the B.S. degree in mechanical engineering from Rensselaer Polytechnic Institute, Troy, NY, in 1986 and the M.S. degree in electric engineering at the University of Texas at Arlington in 1993. He is currently pursuing the Ph.D. degree in computer and systems engineering at Renselaer on a U.S. Dept. of Energy Fellowship for Integrated Manufacturing for work on aluminum extrusion process control. He worked for General Dynamics, Fort Worth Division/Lockheed Fort Worth from May 1986 to January 1994. During this period, he worked on many aspects of flight control hardware and system architecture on a variety of concept and prototype aircraft, 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 (S’79–M’81–SM’93) received the B.Eng. degree from McGill University, Montreal, Canada, in 1979, the M.S. degree from the University of Illinois, Urbana-Champaign, in 1981, and the Ph.D. degree from Rensselaer Polytechnic Institute, Troy, NY, in 1985, all in electrical engineering. From 1981 to 1983, he was a System Engineer with Fisher Control Company, Marshalltown, IA, where he worked on the coordination control of a pulp 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 and multiple-robot coordination. Since 1988, he has been with the Department of Electrical, Computer, and Systems Engineering at Rensselaer Polytechnic Institute, where he is currently a Professor. He is also a member of the New York Center for Advanced Technology in Automation, Robotics and Manufacturing. His current research interests are in the area of modeling and control of multibody mechanical systems, including flexible structures, robots, and vehicles, and material processing systems such as extrusion and resistive welding.