Inference-Based Scheme for Controlling Product End-Use Properties

Ind. Eng. Chem. Res. 2010, 49, 8021–8034

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Inference-Based Scheme for Controlling Product End-Use Properties in Reactive Extrusion Processes S. C. Garge,† M. D. Wetzel,‡ and B. A. Ogunnaike*,§ ExxonMobil Corporation, Fairfax, Virginia, 22037, E. I. du Pont de Nemours and Co., Inc., Wilmington, Delaware 19880, and Department of Chemical Engineering, UniVersity of Delaware, Newark, Delaware 19716

Reactive extrusion processes are typically multivariable, display highly nonlinear characteristics, and often have significant time delays associated with the (offline) measurements of key product properties. Achieving desired product characteristics in industrial practice has therefore been based primarily on the control of a single critical variable such as viscosity. However, increasingly stringent customer demand on product quality has rendered such strategies no longer viable and has necessitated the development of more comprehensive schemes that focus explicitly on controlling product quality characteristics. This paper reports on an experimentally validated inference-based control scheme for controlling product quality and end-use properties in reactive extrusion processes. The scheme employs inference models to predict infrequently measured properties at a much faster ratespredictions that are then used to take necessary control action in between samples, within a cascadelike structure involving separate and distinct multivariable controllers. The control scheme is evaluated first in simulation and then implemented experimentally via a Labview-Matlab interface on an actual pilot-scale reactive extrusion process, where product viscosity, tensile strength, and toughness are controlled simultaneously. Some representative results are presented to highlight the advantages and limitations of the scheme. 1. Introduction Reactive extrusion processes have become important in the polymer industry because of their wide range of applications in manufacturing pure polymers, blends, and, more recently, nanocomposites. In a typical reactive extrusion process, the synthesis of the polymeric material (which often involves reaction with other polymers) is carried out in conjunction with processing into a finished product.1 Because the reaction is usually preceded by melting of the solid feed materials in the extruder, the overall process is highly nonlinear due to complex interactions between melting, reaction, and mixing in the convoluted screw sections of the extruder.2 Apart from the nonlinear process characteristics, one of the major challenges associated with the control of reactive extrusion processes is that the product end-use properties, “w” (for example, toughness, tensile strength, etc.), are usually measured offline at a slow and variable rate. To compensate for the nonavailability of end-use property measurements at a frequent enough rate, the product quality variable, “q” (e.g., melt index, viscosity, density, etc.) is measuredsin some cases even onlinesat a relatively faster rate and used for control. The process variables, that is, the manipulated inputs, “u” (screw speed, feed rate, etc.), and the process outputs, “y” (die pressure, melt temperature, motor power, etc.), are measured at a much higher frequency with almost no delay. Most recent attempts at controlling reactive extrusion processes have focused on the control of process outputs or one of the product quality variables,3,4 but this is inadequate to guarantee that the desired end-use property specifications will be met. To meet customer demands efficiently and guarantee acceptable end-use product performance, it has become impor* To whom correspondence should be addressed. E-mail: ogunnaike@ udel.edu. † ExxonMobil Corporation. ‡ E. I. du Pont de Nemours and Co., Inc. § University of Delaware.

tant to develop techniques for explicit control of not only the process outputs y and the product quality variables q, but, more importantly, the end-use properties w. In this paper, we present such a control scheme for controlling the product end-use properties explicitly, in order to ensure acceptable end-use performance. The paper is organized as follows: the main issues associated with the control of reactive extrusion are discussed in section 2 before various components of the proposed modeling and control scheme are described in section 3. The procedure for designing and implementing the controllers on an experimental system is then discussed in sections 4 and 5 along with some simulation results. Finally, the experimental evaluation of the control scheme on a pilotscale process is presented in section 6. 2. Control Relevant Considerations The following physical characteristics of a reactive extrusion process strongly influence the evolution of final product properties and are therefore essential to the effective control of these properties: (1) Nonuniform feed material properties often result in undesirable variability in product properties, leading to unacceptable product end-use performance. (2) Poor heat transfer characteristics lead to nonhomogenous reaction rates for temperature-dependent reactions. (3) Product morphology, which is mainly developed during the polymer melting process, is an important determinant of product end-use properties. If significant reaction occurs as polymers melt, the reaction can also have a pronounced effect on the product morphology, making the control of product properties even more challenging. (4) Die flow instabilities (e.g., sharkskin effect), melting instabilities, and the periodic nature of feeder screws are additional factors that contribute to variations in product properties.5

10.1021/ie100435s  2010 American Chemical Society Published on Web 05/20/2010

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Ind. Eng. Chem. Res., Vol. 49, No. 17, 2010

Table 1. Literature Review on “Modeling of Reactive Extrusion” a

ref reactants 6 18 19 20 21 22

CL PP N PP BMA U, IC

b

initiator TPT FR PO FR

reaction model model typec natured typee LP D PC D P P

T SS SS SS SS SS

CS KM PF CS CS AD

predicted variablesf DF/P/T/ η/RTD/X MWD Η DF/P/T/ η/RTD/X MW, η RTD

a

Polypropylene (PP), ε-caprolactone (CL), nylon (N), butyl methacrylate (BM), urethane (U), isocynates (IC), low density polyethylene (LDPE), ethyl methacrylic acid (EMMA), ionomer (I). b Tetrapropoxy titanium (TPT), peroxide (PO), radical initiators (RI). c Degradation (D), living polymerization (LP), polycondensation (PC), polymerization (P), neutralization (N). d Steady state (SS), transient (T). e Tank in series (TS), plug flow (PF), kinetic-melting (KM), axial dispersion (AD). f Degree of fill (DF), pressure (P), temperature (T), viscosity (η), residence time distribution (RTD), conversion (X), molecular weight (MW), molecular weight distribution (MWD), polydispersity index (PDI).

To control key product properties effectively, it is necessary to understand and quantify the role of the process physics that are responsible for these key characteristics. However, for most reactive extrusion processes, the process physics are unknown a priori, which makes it difficult to build reasonably accurate fundamental process models. Table 1 shows a summary of published reactive extrusion process models. By far the most attention has been paid to building steady-state process models, which are useful for process design and optimization but not for process control: inferring product properties dynamically, a requirement for effective control, is not possible with such models. The one notable exception6 is a dynamic process model for predicting product composition and pressure profile dynamics. Nevertheless, even for those reactive extrusion systems for which the process physics have been studied extensively, the role of these process physics in the evolution of product quality and end-use properties is still not completely clear. The fundamental complexity of reactive extrusion process characteristics manifests in different ways and presents a wide variety of challenges to the development of an effective scheme for controlling product end-use properties. First, these fundamental characteristics are directly responsible for the observed highly nonlinear behavior; they also contribute to rampant process disturbances (e.g., flow instabilities). Second, these complex process characteristics make it very difficult to obtain high fidelity predictions of infrequently measured product quality and end-use propertiessa major challenge to any scheme based on inference models. It is not surprising, therefore, that a review of recent literature on reactive extrusion control schemes, presented in Table 2, reveals an emphasis on the control of single product quality attributes such as viscosity, with no attempts to date at tackling the fundamental problem of controlling product end-use properties directly. (Note that this review does not include the control of food extrusion processes, since food extrusion has entirely different process characteristics and control objectives compared to those in polymer extrusion.) 3. Modeling and Control Scheme In the broadest sense, the proposed paradigm for controlling the product quality and end-use properties of a reactive extrusion process is predicated upon developing quantitative relationships between variables across the entire processing chainsconsisting of the reactive extrusion process along with such postprocessing

Table 2. Literature Review on “Process Control of Reactive Extrusion” ref reactantsa initiatorb

reaction typec

control schemed IMC OC, CMV, PP, MV, PID PI, MV RC

23 CL 3, 24 PP

TPT PO

P D

4 25

TPT

N P

EMMA, I CL

manipulated controlled variablese variablesf IP IC

Η Η

FR IP

Η P

a Polypropylene (PP), ε-caprolactone (CL), nylon (N), butyl methacrylate (BM), urethane (U), isocynates (IC), low density polyethylene (LDPE), ethyl methacrylic acid (EMMA), ionomer (I). b Tetrapropoxy titanium (TPT), peroxide (PO), radical initiators (RI). c Degradation (D), living polymerization (LP), polycondensation (PC), polymerization (P), neutralization (N). d Internal model control (IMC), optimal control (OC), minimum variance (MV), constrained minimum variance (CMV), pole placement (PP), proportional integral derivative (PID), robust control (RC), proportional integral (PI). e Initiator composition (IC), feed rate (FR), inputs (IP). f Pressure (P), viscosity (η).

operations as quenching, injection molding, etc.sand thereafter utilizing these relationships for three important tasks: (i) to infer infrequently measured product quality variables and end-use properties at a reasonably fast rate; (ii) to translate the customer requirements on end-use performance to set points for the output variables, y; and (iii) to make appropriate adjustments (i.e., control action) wherever appropriate along the processing chain, based on all available information. To guarantee acceptable enduse performance, it will be necessary to extend the paradigm to include customer feedback, z, a binary variable that is 1 for acceptable performance and 0 for unacceptable performance. However, such an extension lies outside the intended scope of the current work. Nevertheless, to implement the complete paradigm, we propose a multivariable cascade-type control scheme in Figure 1, where C1 is a fast model-based controller for rejecting rampant unmeasured disturbances and C2 is a slower model-based controller for translating end-use performance objectives to set points for the process outputs, y. In addition to these controllers, basic regulatory controllers (not shown explicitly) are used in traditional fashion to ensure that the set point changes in the manipulated variables, u, are efficiently tracked. Since the product quality and end-use properties are measured at an unacceptably slow rate, predictions obtained from high-fidelity inference models are used in the C2 controller loop as surrogates for the infrequently available measurements. The indicated third loop shows C3, a controller for translating customer feedback into end-use performance objectives. The development and implementation of this loop is reserved for future work but is indicated here for completeness; the scope of this paper is restricted to the design and implementation of the controllers C1 and C2. 3.1. Models for Controller Design. A schematic representation of the proposed modeling scheme is shown in Figure 2; it consists of the following models required for designing the multivariable controllers: (i) Muysa model relating the manipulated variables, u, to the process output variables, y; (ii) Myqwsa model relating the process outputs, y, to the product quality variables, q, and the end-use properties, w; and (iii) Mwzsa model relating the end-use properties to the customer feedback, z (not discussed further in this paper). While the model Muy has been an inseparable component of most model-based control schemes, the models Myqw and Mwz are rarely (if ever) employed in such schemes for the obvious reasons that typical modelbased control schemes have not been concerned with end-use characteristics or customer feedback. 3.2. Models for Inference. The process physics that relate the process outputs (y) in a reactive extrusion process to the


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Figure 1. Inference-based scheme for controlling product properties of a reactive extrusion process.

Figure 2. Modeling scheme consisting of models required for designing the controllers in the proposed control scheme.

Figure 3. Inference modeling scheme consisting of models required for predicting infrequently measured product properties.

product quality (q) and end-use properties (w) are often unknown. As a result, it is difficult to develop the model Myqw from first-principles. Moreover, since the measurements, y, are usually susceptible to rampant process disturbances, it is seldom possible to develop empirical relationships that will be sufficiently useful for inferring properties with reasonable precision and accuracy. It is necessary, therefore, to find alternative means for building inference models of this type that can provide reliable estimates of end-use properties. Such an alternative is shown in Figure 3, a schematic representation of the proposed inference scheme consisting of the following models: (i) Muq′srelating the manipulated variables, u, to the “internal” product quality variables, q′; (ii)

Mq′qsrelating the internal quality variables, q′, to the product quality variables, q; (iii) Mumsrelating the input variables, u, to the melting process parameters, m; and (iv) Mmwsrelating the melting process parameters to the end-use properties, w. Notice that we have introduced two additional classes of variables: (i) internal product quality variables, q′ (typical examples are composition and weight-average molecular weight); and (ii) melting process parameters, m (typical examples are melting process gain and time constant). Even though the internal product quality variables are not measured, they can be accurately predicted using a fundamental process model (Mqq′) and are strongly correlated to the measured product quality variables, q. In many reactive extrusion processes, the melting process plays the most dominant role in the evolution of such product end-use properties as tensile strength and toughness. Predicting these properties accurately, therefore, requires quantifying the role of the melting process appropriately. However, because the physics of the melting process are not well-known, it is difficult to build a fundamental, physics-based melting model. As an alternative, we have found that end-use properties are strongly correlated with the melting process parameters, parameters that can be determined fairly straightforwardly by the simple procedure described in refs 7 and 8. Taking advantage of this fact, a two-step method is proposed for quantifying the role of the melting process. First, a correlation between the process inputs, u, and the melting process parameters, m, (model Mum) is developed, following which the model Mmw relating the melting process parameters to the end-use properties is then developed. While the detailed procedure for developing these inference models for the example pilot scale reactive extrusion process used in this study has been presented elsewhere,7-10 some of the salient points are described below: 1. Model Muq′. • This model, which quantifies the effect of dynamic changes in the manipulated inputs, u, on the internal quality variables, q′ (composition, molecular weight, etc.), is partitioned into two: a flow model and a reaction model. • The flow model is composed of an algebraic residence time distribution (RTD) model, which describes the residence time distribution as a function of the inputs, u, along with a “degree of fill profile” model, which describes the degree of fill profile of the polymer melt in the extruder.

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• The reaction model quantifies the effect of the reaction on the product composition and average molecular weight (q′). The extruder is modeled as a nonisothermal axial dispersion reactor. 2. Model Mq′q. • A modified Carreau model11 is used to relate the molecular weight (q′) to the melt viscosity (q). 3. Models Mum and Mmw. • These models are developed from experimental data obtained using response surface methodology. The statistical experimental design procedure is used to obtain optimally informative data in order to approximate the relationship between the independent factors and the measured responses with the “best” linear or quadratic curve. For instance, the factors selected for developing the model Mmw are the melting model parameters (m) and the selected responses are the end-use property measurements (w). This method yielded the following linear model: w ) c0 +

∑cm i

i

(1)

with ci as the regression constants and mi as the melting process parameters such as process gain and time-constants.8,10 4. Design and Implementation of Controller C1 4.1. Experimental Reactive Extrusion Process. The experimental system employed in this study involves the reaction of a functionalized ethylene terpolymer, “Elvaloy” (ethylene/ n-butyl acrylate/glycidal methacrylate terpolymersEBAGMA) with an acid copolymer “Nucrel” (ethylene/methacrylic acid copolymersEMAA), in a Coperion W&P ZSK-30 mm corotating, intermeshing twin-screw extruder. The extruder is equipped with a data acquisition system (DAQ) described in detail by Wetzel et al.12 The following process signals are accessible through the DAQ hardware: screw speed (u1), EMAA feed rate (u2), EBAGMA feed rate (u3), die pressure (y1), and motor power (y2). The final process output, feed EBAGMA weight fraction (y3), is calculated from the feed rates of the two polymers according to y3 )

u3 u2 + u 3

(2)

4.2. Process Model Identification. The empirical process model, Muy, which relates the manipulated inputs, u, to the process outputs, y, was identified from input-output data. First, because the process is nonlinear, we performed preliminary experiments to obtain a priori process knowledge necessary for designing suitable identification tests (for example, process gains, dominant time constants, etc.). These experiments consisted of carefully designed dynamic tests at two operating points: (A) feed EBAGMA weight fraction, y3 ∼ 0.01 and (B) feed EBAGMA weight fraction, y3 ∼ 0.04. Since EBAGMA was the limiting reactant, the EBAGMA feed composition determined the reaction rates in the extruder. Independent step changes in the manipulated inputs were used to obtain the process gains (Table 3) and time constants, whereas simultaneous staircase changes were implemented to assess the extent of process nonlinearity around each operating point. The experiments provided the following process insights (first presented in Garge et al.9): • The process gain, the ratio of the change in the output y to that in the input u, differed significantly at the two operating

Table 3. Scaled Gains and Condition Number (K) at the Two Process Operating Points operating point A

operating point B

O/I

u1

u2

u3

u1

u2

u3

y1 y2 y3 κ

-0.0617 0.75 0

0.0711 0.333 0.0205 185

1.5 0.625 0.4886

0.16 0.381 0

-0.3 0.3 0.0208 8

1 0.188 0.4579

points, and, in some cases, had opposite signs, indicating a strong nonlinear effect of reaction on process dynamic behavior. • Singular value decomposition (SVD) of the scaled gain matrix showed a condition number, κ, (the ratio of the highest to the lowest singular value) that was much higher at the operating point A than at the operating point B. These κ values suggest that the process is ill-conditioned at the operating point A and that the reaction improves the conditioning of the process. • The process is approximately linear in the proximity of each operating point. In general, the proper design of a final identification test depends intrinsically on the desired model form, which is postulated using process information gathered from the preliminary tests. In the specific case under consideration here, the results of the preliminary tests indicate that the process exhibits significant nonlinearity. However, rather than attempt to represent the process behavior over the full range of operation with a single nonlinear model, such as a NARMAX, or a Hammerstein model,13 we opt for a representation based on a series of local linear models. This approach is particularly suitable for our specific process because it is approximately linear at each operating point. Thus, the final process identification exercise consisted of designing and administering appropriate input excitation signals for identifying a linear model at each operating point. Such input signals should be persistently exciting to guarantee that the identified model parameters are unique (see, for e.g., the work of Zhu14). In this study, the generalized binary noise (GBN) signals, proposed by Tullenken,15 were selected as the excitation signals because they are persistently exciting and are easy to administer. Specially designed signals such as the GBN signals have one advantage over conventional step test signals: the test durations are much shorter because these signals can be administered simultaneously for multiple inputs. A GBN signal switches between the values of -a and a (a being the magnitude of the desired change) using a switching probability, psw, calculated according to the following rule: P[u(t) ) -u(t - 1)] ) psw P[u(t) ) u(t - 1)] ) 1 - psw

(3)

As recommended by Zhu,14 GBN signals with a mean switching time of 120ssequal to one-third of the dominant process time constantswere used in the final test. In recognition of the presence of unavoidable unmeasured disturbances, and because of high measurement noise levels (resulting in low signal-to-noise ratios), the test duration was set at approximately 15 times the dominant time constant. Figure 4 shows the uncorrelated GBN signals administered simultaneously in the inputs, screw speed (u1), EMAA feed rate (u2), and EBAGMA feed rate (u3), at each operating point. Since the process is ill-conditioned at the operating point A, the uncorrelated signals are not suitable for use in identifying the low gain direction. An additional test, based on the open loop


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Figure 4. Final identification test results (at operating point A) using uncorrelated GBN input signals.

Figure 5. Final identification test results (at operating point A) using partially correlated GBN input signals to identify the dynamics in the low gain direction.

design suggested by Zhu,14 was used to identify the low gain direction of the process at the operating point A. The input signals, shown in Figure 5, have correlated high amplitude periodssrequired to identify the low gain directionscombined with uncorrelated low amplitude periods. 4.3. Model Structure and Order Selection. The following set of candidate parametric models was considered for each operating point: (1) Autoregressive with exogenous inputs (ARX), where disturbances are modeled as white noise and are a part of the system dynamics. (2) Autoregressive moving average with exogenous inputs (ARMAX), where disturbances are modeled separately from system dynamics and enter as disturbances to the input. (3) Box-Jenkins (BJ), suitable when disturbances enter as “process” disturbances.

The general (single-input-single-output: SISO) structure of these parametric models is as follows F(q)y(t) )

B(q) C(q) u(t - nd1) + e(t - nd2) A(q) D(q)

(4)

where, A(q) ) 1 + a1q-1 + ... + anq-na B(q) ) b1q-1 + ... + bnq-nb

C(q) ) 1 + c1q-1 + ... + cnq-nc

D(q) ) 1 + d1q-1 + ... + dnq-nd and, q-kV(k) ) V(t - k)

Parameter estimation was performed using the system identification toolbox of Matlab, using the following criteria for comparing the model structures and selecting appropriate model

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Table 4. Best-Fit BJ Model Orders at the Two Process Operating Points O/I

u1 (na, nb)

u2 (na, nb)

u3 (na, nb)

noise (nc, nd)

operating point A y1 y2 y3

5, 5 4, 2 2, 2

y1 y2 y3

4, 4 3, 3 2, 2

5, 5 4, 2 2, 2

5, 5 4, 3 2, 2

2, 2 1, 1 1, 1

4, 4 3, 3 2, 2

3, 3 1, 1 1, 1

operating point B 4, 4 3, 3 2, 2

orders: (i) percentage of output variation reproduced by the model, (ii) Akaike’s final prediction error, and (iii) pole-zero diagrams to check for overparameterization. Among the candidate models, the multi-input-single-output (MISO) BJ model form provided the best fit and the smallest final prediction error (See the work of Garge9 for full details.). Table 4 shows the best fit MISO BJ model orders at the two operating points, and Figure 6 shows a comparison between the experimental data and model predictions. Reactive extrusion processes are affected by both input and process disturbances. While input disturbances are typically due to compositional and property variations in the polymer feed, process disturbances arise from flow instabilities, the sharkskin effect, etc. However, the fact that the BJ model proved to be more appropriate indicates that for the specific experimental system we have used, process disturbances have a more pronounced effect on the dynamics than input disturbances. 4.4. Controller Design. The purpose of the inner loop controller C1 is to eliminate process and input disturbances and track the set-points specified by the outer loop controller C2. This inner loop controller is implemented as a model predictive controller with the following properties: (1) A sampling interval was chosen as 10 s based on the natural “process response time” estimated from the set of preliminary experiments. (2) The process output predictions are based on the empirical model Muy implemented in discrete state-space form: x(k + 1) ) Ax(k) + Bu(k) + V1(k) y(k) ) Cx(k) + Du(k) + V2(k)

(5)

Here, as usual, x is the vector of system states, and the vectors V1 and V2 represent process and measurement noises respectively. (3) The discrete control action is calculated by solving the following optimization problem at each sampling instant, k: p

min

u(k|k)...u(k+m-1|k)

∑ [y

Qyk+l + uk+lTRuk+l]

T k+l

subject to the following constraints:

[

(6)

l)1

][

]

umin(k) u(k|k) umin(k + 1) u(k + 1|k) e e l l umin(k + p - 1) u(k + m - 1|k) umax(k) umax(k + 1) (7) l umax(k + m - 1)

[

]

[

][

]

ymin(k) y(k|k) ymin(k + 1) y(k + 1|k) e e l l ymin(k + m - 1) y(k + m - 1|k) ymax(k) ymax(k + 1) (8) l ymax(k + m - 1)

[

]

where Q and R are diagonal weighting matrices, p is the prediction horizon, and m the control horizon. (4) The controller selects either of the two linear model components to use for future output predictions based on the feed EBAGMA weight fraction (y3). Thus, the overall operating space is divided into the following two regions: (I) 0 < y3 < 0.025: where the linear model identified at the operating point A is used for the prediction; and (II) 0.025 < y3 < 0.05: where the linear model identified at the operating point B is used. (5) As in many standard MPC formulations, it is assumed that the difference between the model predictions and the process output measurements is due to output step disturbances that remain constant into the future. The estimates of the output disturbances are therefore included in the future output predictions, to incorporate integral action. 4.5. Controller Tuning Considerations. The performance of this controller was evaluated first in simulation to ensure that the controller is tuned properly for satisfactory set-point tracking and disturbance rejection. The main challenge to satisfactory controller tuning is the nonlinear and ill-conditioned nature of the process. The safe process operating limits shown in eqs 7 and 8, especially the ones imposed on the screw speed manipulation to ensure smooth running of the screw motor, play a major role in determining controller aggressiveness. In addition, to prevent the system from becoming unstable as a result of rapid switching between the two operating regions, the output y3 needs to be appropriately weighted in the controller tuning process. The performance was evaluated in simulation under the following conditions, using a “plant model” obtained by multiplying corresponding state-space matrices of the prediction model by 0.95 to introduce plant-model mismatch: (i) Setpoint changes were implemented at t ) 0; die pressure, y1, from 280 to 305 psi; motor power, y2 from 1700 to 1850 W; feed EBAGMA weight fraction, y3 from 0.025 to 0.045. (ii) Simultaneous step change disturbances were implemented at t ) 0.4 h; y1 -10 psi; y2 -25 W. (iii) The controller tuning parameters were chosen as follows: Q ) diag[0.1 0.1 12 000]; R ) diag[0 0 0]; prediction horizon )500 s; control horizon ) 250 s. The simulation results are shown in Figure 7, where we observe that while the controller was able to track the set-points for y2 and y3 effectively, it was unable to eliminate the offset in y1. The reason for this offset is as follows: while, in principle, there are three manipulated variables, u1, u2, and u3, to be used to control three “output” variables, y1, y2, and y3, in actual fact, y3 is a ratio of u3 to (u2 + u3), so that there are only two truly independent manipulated inputs. With the loss of one degree of freedom, only two of the three output variables can be driven independently to arbitrary set-points. Now, because the two operating points around which the empirical process models were developed are characterized by values of y3, a significant penalty was placed on deviations from the desired set-point for


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Figure 6. MISO Box-Jenkins model fit to the output (y) data collected in response to GBN input signals. In the right-hand panels, the solid lines represent the model predictions; the marked lines represent the output measurements.

this variable (see tuning matrix Q) in order to prevent the controller from switching aggressively between the two linear prediction models. The choice of the weighting matrix Q therefore ensured that y2 and y3 will achieve their respective set-points, leaving an offset in y1. Note that achieving the desired set-point for y3 implies that u2 and u3 must change together as dictated by the desired ratio. This latter fact is confirmed in the trajectories of the input variables shown in the left-hand panel of Figure 7. We note, finally, that, in practice, an offset in y1 will not affect the performance of the overall control scheme significantly because the outer loop controller C2, as is the case with standard cascade controllers, will compensate for any such inner loop offsets. Nevertheless, there are also process-related reasons for tuning the controller C1 as we have done here. Specifically, y3 has a much stronger effect on product end-use properties than does y1, while, in addition, y1 is more prone to rampant process disturbances. Thus, more emphasis should be given to tracking the y3 set-points specified by the controller C2 than tracking the y1 set-points.

5. Design and Implementation of Controller C2 5.1. Controller Design. The main function of the controller C2 is to enable the achievement of targets for product quality (q) and end-use properties (w) by manipulating the set-points for the process outputs (y), which in turn are to be tracked by the inner-loop controller C1. The illustrative experimental process used in this paper involves one product quality variable, viscosity (q) and two end-use properties, tensile strength (w1) and toughness (w2). For this process, the controller C2 determines appropriate set-points for the three process outputs: die pressure, motor power, and feedcomposition. A model predictive controller with the following features is designed for this purpose: (1) Control action is calculated at discrete points in time, with a sampling interval of 200 s. The choice of the sampling interval is dictated by the response time of the inner control loop. (2) Product quality and end-use characteristics predictions are based on the linear model Myqw according to

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Figure 7. Controller C1 performance: simulation results for set point tracking and disturbance rejection objectives. The dashed lines in the right-hand panels represent the desired set-points, and the marked lines represent the process output (y) values.

w ) K wy q ) Kqy

(9)

Here, Kw is a 2 × 3 steady state gain matrix and Kq, a 1 × 3 vector, whose specific values, obtained from experimental data discussed in the work of Garge et al.,8 are

[

-3 4.8313 × 10-4 4.7083 Kw ) -2.3859 × 10 -0.12905 0.014017 170.87 Kq ) [5.5942 0.46050 4.3578 × 104 ]

] (10)

(3) Control action is calculated at each sampling instant, k, by solving the following optimizing problem: p

min

y(k|k)...y(k+m-1|k)

∑ [ω

Qωk+l + yk+lTRyk+l] (11)

T

k+l

l)1

subject to the following constraints:

[

][

]

ωmin(k) ω(k|k) ωmin(k + 1) ω(k + 1|k) e e l l ωmin(k + p - 1) ω(k + m - 1|k) ωmax(k) ωmax(k + 1) (12) l ωmax(k + m - 1)

[

]

[

][

]

ymin(k) y(k|k) ymin(k + 1) y(k + 1|k) e e l l ymin(k + m - 1) y(k + m - 1|k) ymax(k) ymax(k + 1) (13) l ymax(k + m - 1)

[

]

where ω ) [q w1 w2]T; y ) [y1 y2 y3]T; ω ˜ are the product property targets; Q and R are the diagonal weight matrices; p is the prediction horizon, and m is the control horizon. (4) Inference models (Muq′, Mq′q, Mum, Mmw) are used as surrogates for the “measurement sensors” to provide estimates of the product quality and end-use properties in between the infrequent measurement samples. (5) The difference between the model Myqw predictions and the measurements/estimates of the controlled product properties is assumed to be due to output disturbances, which are included in the future output predictions as steps, providing integral action in the controller C2. (6) Any offsets resulting from the conservatively tuned controller C1 are accounted for in future output predictions as input disturbances. 5.2. Simulations: The Role of Process Physics. The performance of the overall control scheme was evaluated in simulation under conditions shown in Table 5. Beginning from Table 5. Product Property Baseline and Set-Points Used in the Simulations

baseline simulation I: set-points simulation II: set-points

viscosity, q (Pa s)

tensile strength, w1 (MPa)

toughness, w2 (J/m3)

1220 1420 1020

8.88 9.38 9.38

142.44 152.44 152.44


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Figure 8. Controller C2 performance: response under conditions listed in Table 5. The dashed lines represent the (y/q/w) set-points, marked lines represent the product property values, and the solid lines represent the process outputs (y), which are controlled by controllerC1.

the baseline conditions, in simulation I, simultaneous step increases are made in the desired set-point values of the product quality variable, q (viscosity), and in the two end-use properties, w1, (tensile strength), and w2, (toughness) as indicated. In simulation II, from the same baseline conditions, this time the viscosity set-point is set below the baseline simultaneously with

step increases in the set-points for the two end-use properties that are of the same magnitude as those in simulation I. Thus, the desired conditions for the two simulations are the same except for the singular (and very important) fact that the viscosity target is set above the baseline in the first simulation and below the baseline in the second. The inference models

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Figure 9. Labview-Matlab interface for experimental implementation of the control scheme on a pilot-scale reactive extrusion process. Table 6. Product Property Baseline and Set-Points in the Experiments % PE in EBAGMA feed baseline experiment I

0 20

experiment II

50

comment original new baseline set-point new baseline set-point

tensile viscosity strength toughness (Pa s) (MPa) (J/m3) 800 720 800 690 800,725

8.88 8.88 8.88 8.88 8.88

142.44 142.44 142.44 142.44 142.44

(Muq′, Mq′q, Mum, Mmw) were used to obtain estimates of the product quality and end-use properties, with plant/model mismatch incorporated into the inner control loop as stated in section 4.5. The prediction horizon was set at 6000 s, the control horizon at 3000 s. With the inner loop controller C1 implemented as designed in the previous section, the overall control system performance in simulation I is shown in Figure 8a, where the tuning parameters for controller C2 are the following: Q ) diag[1 1 1]; R ) diag[1 2 2]. The right-hand panel shows how, by manipulating set-points for the output y variables as indicated in the dashed lines in the left panel, controller C2 is able to track the viscosity set-point accurately, and the end-use property set-points with small offsets. How the controller C1 is able to track these desired output y set-points is shown in the solid lines in the left-hand panel. As expected from the previous section’s discussion of the performance of controller C1, the y3 set-point is tracked perfectly while the other two output variables show steady-state offsets. These simulation results illustrate some fundamental and interesting principles behind effective control of product quality and end-use characteristics. Even when it is an empirical model, the model Myqw, which is central to the implementation of such a control scheme, is a reflection of the physics intrinsic to the relationship between the process outputs and the product quality/end-use characteristics. Regardless of the set-points demanded of the product quality and end-use characteristics, this model provides an indication of what is physically possible and what is not. Thus, for example, the gain matrix for the specific process used in this simulation, whose elements are shown in eq 10, is very poorly conditioned, with singular values σ1 ) 170.9, σ2 ) 5.6, and σ1 ) 0.0001, so that the condition number is 1.7 × 106. Since attaining arbitrary set-points for q and w by manipulating the outputs, y is tantamount to inverting this gain matrix, the implications of this are that if the vector of desired set-points is not in the “natural direction,” it will be essentially impossible to achieve these desired set-points. Such natural directions are dictated, of course, by the physics of the process so that the only achievable setpoints for q and w are ones that are supported by the process physics; anything else will result in significant offsets. Thus,

the small offsets shown in Figure 8a are indicative of the best possible performance given what the physics of the process will allow. The set-points chosen for simulation II were selected to buttress this last point. One of the points demonstrated by simulation I is that demanding simultaneous increases in product viscosity and in the end-use properties is physically meaningful and feasible, to a certain extent. With simulation II, the demand is to have a product with lower viscosity but higher tensile strength (w1) and toughness (w2). This is physically impossible. It is not surprising therefore that, as shown in Figure 8b, the control system in the second simulation was able to decrease the viscosity only up to a certain point while simultaneously increasing the tensile strength and toughness. The final result in the right-hand panel is a large offset in viscosity and relatively smaller ones in toughness and tensile strength. The performance of controller C1 in tracking the inner loop process output setpoints is shown in the left-hand panel, where, once again, by design, the y3 set-point is tracked perfectly. To conclude, we note that the overall control system performance was much better for the objectives set in the first simulation compared to the ones set in the second simulation, primarily because of the intrinsic characteristics of multivariable control of product quality/end-use characteristics: the achievable set of desired product quality/end-use characteristics set-points are determined by the process physics. Well-designed controllers can only provide the best possible physically realizable compromise for other arbitrarily specified set-points. 6. Experimental Implementation 6.1. Hardware-Software Implementation Scheme. Online implementation of the overall control technique on a real process requires real-time data acquisition (in this case using the DAQ hardware), optimal control action computation (in a multirate fashion), and control action implementation (within the control interval of 10 s). Our experimental implementation of the control system on the pilot-scale extruder described in section 4.1 employed the configuration shown schematically in Figure 9, involving a hierarchical scheme similar to that in the work of Lawrence et al.16 in which a “hardware controller” acts as an interface between the mathematical software used for calculating optimal control action and the actual extrusion process. The hardware controller is responsible for acquiring sensor signals from the DAQ, redirecting them to the mathematical software, obtaining computed control action from the software, and converting these into appropriate form (current/voltage/frequency), which is then ultimately sent to the DAQ hardware for implementation on the physical process.


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Figure 10. Controller C2 performance: response under conditions listed in Table 6. The dashed lines represent the (y/q/w) set-points, marked lines represent the product property values, and the solid lines represent the process outputs (y) measurements. The diamond markers represent the offline w measurements performed after the completion of the experiment.

The hardware controller was developed in Labview, while control action calculations were performed in Matlab, allowing us to take advantage of the easy-to-use Labview ActiveX interface with Matlab. Furthermore, the multivariable multiloop control scheme can be implemented elegantly with the “timed” loops in Labview. 6.2. Experiments. The overall control system designed and evaluated in simulation in sections 4 and 5 was implemented on the experimental pilot-scale extruder, and its performance evaluated under conditions shown in Table 6. High-density polyethylene (PE) was added to the EBAGMA base feed before the start of each experiment to introduce a change in base-level

feed-material properties. The presence of PE in the feed mixture suppresses the cross-linking reaction between the feed polymers, causing a reduction in the viscosity and thereby introducing a significant step disturbance. The experimental investigations involved restoring the product quality/end-use characteristics variables to their respective original desired set-point values. Since there is usually no control over the quality of the feed polymers (which is often prone to significant variability), feed composition disturbances constitute one of the most common classes of disturbances experienced by reactive extrusion processes.

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Figure 11. Follow-up controller C2 simulation results: response under conditions of experiment II. The diamonds represent the online w measurements, which were used for updating the inference model Mmw.

The controllers C1 and C2 were implemented as designed in the preceding sections, with the models that were used for the simulations in the previous section now serving as the prediction models. (Viscosity measurements were available online with a small delay of 180 s, making viscosity predictions from the inference models Muq′ and Mq′q unnecessary.) As in commercial operations, tensile strength and toughness measurements were unavailable during the experiments. Instead, product samples were collected intermittently during the second set of experiments and were later compression-molded and tested on an Instron universal test machine (according to ASTM D638) to determine their tensile strength and toughness. Experiment I. The objective of this experiment was to restore the viscosity to its original baseline value of 800 Pa s while maintaining tensile strength and toughness constant. As shown in Figure 10a, the overall control system performed this task successfully. The right-hand panel indicates how controller C2 restored the viscosity to the desired set-point in under 30 min and maintained the end-use properties essentially constant at the respective desired values. Note that with the exception of viscosity, these are simulation results since product end-use characteristic data were not available during this experiment. The left-hand panel, on the other hand, shows actual experimental data. It demonstrates how the process output variables tracked the series of set-point changes required to be tracked by controller C1 in the inner loop in order to achieve the product quality/end-use characteristics results shown in the right-hand panel. As expected, the set-point changes in y3 were tracked much better than the set-point changes for y2. The die pressure measurements, y1, are noisy and were affected much more severely by process disturbances due to periodic accumulation and discharge of the melt in the extruder end zone. The setpoint changes in this variable were therefore more difficult to track compared to y3. Experiment II and Follow-up Simulation. Due to the higher quantities of PE in the feed in this second set of experiments, the disturbance effect is greater. The objective in this experiment is twofold: (i) to restore the product

quality/end-use characteristics variables to their respective original set-point values and (ii) to track a set-point change in viscosity (from 800 to 725 Pa s) implemented at 4200 s. The overall system performance is shown Figure 10b. The left-hand panel shows the performance of the inner loop controller, C1, in tracking the series of set-point changes required to achieve the results shown in the right-hand panel. Even though the disturbance to the process is more severe with this set of experiments, the performance of the inner loop controller is comparable to that observed in the first experiment. The right-hand panel of Figure 10b shows how controller C2 was able to return the product viscosity close to its original baseline value and also track the negative setpoint change made at 4200 s. The diamond symbols and the accompanying error bars (95% confidence intervals) show the results of the tensile strength and toughness tests carried out on product samples collected at the beginning, middle, and end of the experimental run. These results, which were obtained after the completion of the experiments, reveal a significant offsets between the actual values and the desired set-points. The primary reason for these offsets is that the effect on the product properties of the PE addition is that of an unmeasured disturbance that is not adequately reflected in the inference models Mum and Mmw. And since these measurements were not available during the course of the experiments, these effects could not be compensated for by feedback to controller C2. This latter point motivated us to perform a follow-up simulation, to assess how the controller would have performed had the lab measurements of the end-use properties been made available each time, albeit after a delay of 30 min. The initial steady state conditions and the set-point objectives were taken to be the same as in experiment II. The product characteristics data, when available, were used to update the inference model Mmw parameters ci (eq 1) online, using the projection algorithm,17 i.e.,


Θnew ) Θold -

Φ [w - Φ′Θold] (Φ′Φ) m

(14)

where Θ is the vector of the parameters ci, with the subscript old referring to the old parameter values and the subscript new refering to the updated parameters values; Φ is the vector of the melting parameters mi, and wm is the measured value of the end-use property, w. The results of this simulation are shown in Figure 11, which demonstrates, as expected, that with online updates controller C2 is able to bring the end-use properties closer to the desired set-points, although at the cost of larger excursions in viscosity. These simulation results illustrate what is more likely to occur in industrial practice, when product properties measurements, even though infrequent, will be available periodically. The performance of the inference-based control scheme will improve when such measurements are used whenever available for correcting the inference model predictions as we have demonstrated here. 7. Summary and Conclusions We have presented an inference-based control strategy for effective control of product quality and end-use properties in extrusion processes and demonstrated its performance both in simulation and on an actual pilot-scale extruder. The salient features of the control strategy are as follows: (i) It takes advantage of the availability of fast process measurements to compensate for the effect of rampant process disturbances on the product properties. (ii) It employs inference models as surrogates for various aspects of the physical process to provide fast, online estimates of the infrequently measured product quality and end-use properties. This facilitates the implementation of control action at a rate faster than the rate at which these properties can be measured. (iii) It employs a multivariate cascadelike structure in which a top-level multivariable controller is designed to enable the achievement of targets on product quality and enduse properties by manipulating the set-points to an inner loop that is under the control of another multivariable controller. (iv) To guarantee acceptable product end-use performance, it will be essential to extend the control scheme to translate customer feedback information into end-use performance objectives. This will be the subject of a subsequent study. By using inference models expressly for predicting product properties, the control scheme enables the usage of a wide variety of inference models that are necessary to provide reliable property predictions. This feature is particularly advantageous for reactive extrusion processes, where detailed process physics are typically not known to the point where a single inference model can be developed for predicting all product properties of interest. The application to a specific example experimental reactive extrusion process provided the following insight into the control of product properties in this important class of processes: (1) The extrusion process was found to be highly nonlinear and ill-conditioned when the extent of reaction is low, making it necessary to use carefully designed tests for process identification. (2) For this specific process, the controller C1 was similar in structure to a gain-scheduled controller that uses a local linear model for future output predictions.

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(3) Two inference models were required for inferring product properties: a fundamental model for melt viscosity predictions and a fundamental/empirical hybrid model for predicting tensile strength and toughness. (4) The implementation of the overall control scheme in simulation revealed that the process physics determine whether or not an arbitrarily specified set of product property targets will be achievable in practice, regardless of the control system design. (5) The experimental implementation indicated that the control scheme is able to track the set-points of frequently measured product quality variables quite effectively; in the absence of feedback from actual measurements, enduse property set-point tracking was not as satisfactory. A follow-up simulation showed that the control scheme performance improved when the feedback from the infrequent measurements was used for improving inference model predictions. Literature Cited (1) Tzoganakis, C. Reactive Extrusion of Polymers: A Review. AdV. Polym. Technol. 1989, 9, 321–330. (2) Janssen, L. On the Stability of Reactive extrusion. Polym. Eng. Sci. 1998, 38, 2010–2019. (3) Pabedinskas, A.; Cluett, W. R. Controller-Design and Performance Analysis for a Reactive Extrusion Process. Polym. Eng. Sci. 1994, 34, 585– 597. (4) Broadhead, T. O.; Patterson, W. I.; Dealy, J. M. Closed Loop Viscosity Control of Reactive Extrusion with an In-Line Rheometer. Polym. Eng. Sci. 1996, 36, 2840–2851. (5) Mudalamane, R.; Bigio, D. I. Process Variations and Transient Behavior of Extruders. AIChE J. 2003, 49, 3150–3160. (6) Choulak, S.; Couenne, F.; Le Gorrec, Y.; Jallut, C.; Cassagnau, P.; Michel, A. Generic Dynamic Model for Simulation and Control of Reactive Extrusion. Ind. Eng. Chem. Res. 2004, 43, 7373–7382. (7) Garge, S. C.; Wetzel, M. D.; Ogunnaike, B. A. An Empirical Model for Melting in a Co-Rotating Twin-Screw Extruder. SPE ANTEC Technical Papers, Charlotte, NC, May 2006. (8) Garge, S. C.; Wetzel, M. D.; Ogunnaike, B. A. Quantification of the Melting Process in a Co-Rotating Twin-Screw Extruder: A Hybrid Modeling Approach. Polym. Eng. Sci. 2007, 47 (7), 1040–1051. (9) Garge, S. C. Development of an Inference Based Control Scheme for Reactive Extrusion Processes. PhD Dissertation, University of Delaware, 2007. (10) Garge, S. C.; Wetzel, M. D.; Ogunnaike, B. A. Modeling for Control of Reactive Extrusion Processes. Proceedings ADCHEM, Gramado, Brazil, June 2006. (11) Carreau, P. J.; De Kee, D. C. R.; Chhabra, R. P. Rheology of Polymeric Systems: Principles and Applications; Hanser Gardner: Cincinnati, 1997. (12) Wetzel, M. D.; Denelsbeck, D. A.; Latimer, S. L.; Shih, C. K. A Perturbation Method to Characterize Melting During the Extrusion of Polymer and Blends. SPE ANTEC Tech. Papers, Chicago, Illinois, May 2004; p 138. (13) Boukhris, A.; Mourot, G.; Ragot, J. Non-linear Dynamic System Identification: A Multi-Model Approach. Int. J. Control 1999, 72, 591– 604. (14) Zhu, Y. C. MultiVariable System Identification for Process Control; Elsevier Science Ltd.: Oxford, 2001. (15) Tulleken, H. Generalized Binary Noise Test-Signal Concept for Improved Identification-Experiment Design. Automatica 1990, 26, 37–49. (16) Lawrence, M.; Erdem, G.; Abel, S.; Morari, M.; Mazzotti, M.; Morbidelli, J. L. Online Optimisation and Feedback Control of a SMB Plant: A LabVIEW Matlab Procedural Implementation. Virtuelle Instrumente in der Praxis, Munich, Germany, October 2004. (17) Goodwin, G. C.; Sin, K. S. AdaptiVe Filtering Prediction and Control; Prentice-Hall: Englewood Cliffs, 1984. (18) Pabedinskas, A.; Cluett, W. R. Balke, S. T.Modeling of Polypropylene Degradation During Reactive Extrusion with Implications for Process-Control. Polym. Eng. Sci. 1994, 34, 598–612. (19) Giudici, R.; doNascimento, C. A. O.; Beiler, I. C.; Scherbakoff, N. Transient Experiments and Mathematical Modeling of an Industrial Twin-

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Screw Extruder Reactor for Nylon-6,6 Polymerization. Ind. Eng. Chem. Res. 1997, 36, 3513–3519. (20) Vergnes, B.; Berzin, F. Peroxide-Controlled Degradation of Poly(propylene): Rheological Behaviour and Process Modeling. Macromol. Symp. 2000, 158, 77–90. (21) Ganzeveld, K. J.; Capel, J. E.; Vanderwal, D. J.; Janssen, L. The Modeling of Counter-Rotating Twin-Screw Extruders as Reactors for SingleComponent Reactions. Chem. Eng. Sci. 1994, 49, 1639–1649. (22) Semsarzadeh, M. A.; Navarchian, A. H.; Morshedian, J. Reactive Extrusion of Poly(urethane-isocyanurate). AdV. Polym. Technol. 2004, 23, 239–255. (23) Gimenez, J.; Boudris, M.; Cassagnau, P.; Michel, A. Control of Bulk Epsilon-Caprolactone Polymerization in a Twin Screw Extruder. Polym. React. Eng. 2000, 8, 135–157.

(24) Pabedinskas, A.; Cluett, W. R.; Balke, S. T. Process Control for Polypropylene Degradation during Reactive Extrusion. Polym. Eng. Sci. 1989, 29, 993–1003. (25) Choulak, S.; Couenne, F.; Thomas, G.; Cassagnau, P.; Michel, A. Methodology for Robust Control of Pressure for Epsilon-Caprolactone Polymerization in a Twin Screw Extruder. Chimia 2001, 55, 244–246.

ReceiVed for reView February 26, 2010 ReVised manuscript receiVed April 30, 2010 Accepted May 3, 2010 IE100435S

Inference-Based Scheme for Controlling Product End-Use Properties

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