Polymer Extrusion: From Control System Design to Product Quality

Article pubs.acs.org/IECR

Polymer Extrusion: From Control System Design to Product Quality Zhijun Jiang,† Yi Yang,*,‡ Shengyong Mo,§ Ke Yao,† and Furong Gao†,§ †

Fok Ying Tung Graduate School, The Hong Kong University of Science and Technology, Clear Water Bay, Kowloon, Hong Kong Special Administrative Region ‡ Department of Control Science and Engineering, Zhejiang University, Hangzhou, China § Department of Chemical and Biomolecular Engineering, The Hong Kong University of Science and Technology, Clear Water Bay, Kowloon, Hong Kong Special Administrative Region ABSTRACT: As a major polymer processing technique, polymer extrusion is a continuous process, during which material properties, machine variables, and process variables interact with each other to determine the final product quality. Precise control of key process variables such as barrel temperatures and melt pressure are crucial to ensure a good product quality in the extrusion process. In this paper, the overall extruder control system is constructed by two independent control loops, a singleinput−single-output (SISO) control of the melt pressure at die output, and a multiinput−multioutput (MIMO) control of the barrel temperatures. The dynamic behaviors of melt pressure and barrel temperatures were analyzed first. The characteristics of the melt pressure dynamics were as follows: nonlinear and time-varying while the extruder barrel temperatures were nonlinear, slow response, and different zones were highly coupled. Advanced control algorithms were adopted to control these key variables. Experimental results demonstrate the fast response, near-zero overshoot, and precise tracking performance of the proposed control strategies. The robustness of the entire control system was verified through different operating conditions including materials and set points. Ultimately the performance of the entire control system was verified by product quality. The product quality improved significantly with the proposed controller.

1. INTRODUCTION The plastic extruder has gained widespread applications in the polymer processing industry for shaping the continuous length of plastic products with constant cross-section, and the products can then be cut into the desired length. Typical products include pipes, sheets, tubes, window frames, electrical wire, cables, and even polymer granules for secondary processes. The machine mainly consists of four components: (1) a feeding section, (2) a barrel screw rotating system, (3) a head with a die to form the desired products, and (4) a controller to control process variables.1 The extrusion process begins by feeding raw material into the hopper in the form of pellets or irregular small bits. Then the polymer granules are conveyed forward by a powerful rotating screw through the heated barrel. As the plastic passes through the screw channel, it is gradually heated and melted by the heat conducted from the heater clipped around the cylinder barrel together with the shear heat generated by the rotating screw. By the time the material is pumped to the tip of the screw it should be well mixed. Then the melted polymer is allowed to pass through a screen pack to filter the contaminants. Finally, it is pushed out of the die with properly regulated temperature and pressure to form the desired products. Then it is pulled by a puller through a cooling system, usually cooling water. Ultimately it is rolled or cut into pieces by the removal. A high extrusion product quality can be indicated by a precisely regulated output volumetric flow. It can be accomplished by controlling the screw revolution speed, melt pressure, and melt temperature within narrow variations. Among all these process variables, melt pressure and melt temperature are the key variables.2 Melt pressure is an important indication of extrusion output rate and process stability. A melt pressure variation of 1% © 2012 American Chemical Society

could lead to an equivalent extruder output variation of 1−3% depending on the rheological behavior of the polymer melt.3 The melt temperature fluctuation influences the material viscosity, density, and degradation kinetics4 and eventually affects the extrusion output rate, product quality, and process stability. Although melt temperature cannot be easily measured, it can be well indicated by barrel temperatures. Therefore, it is extremely important to control the melt pressure and barrel temperatures precisely. In practice, the characteristics of melt pressure are nonlinear and time-varying, moreover with input constraints because the screw revolution speed cannot change rapidly. These characteristics increase the difficulty of melt pressure control. Barrel temperature control is even more complicated: First, the dynamics of the barrel temperature are difficult to model.5 It has an “integrator-like” behavior in the relevant operation range6 compared with melt pressure. The barrel temperatures require a relatively long time for the initial heat up to reach an appropriate operating temperature. Minimizing this time period is important for energy saving and productivity increase. Second, because of the location of the electrical heaters, thermocouples, and the thickness of the cylindrical barrel, the temperature of the heater and the barrel outer wall are higher than that of the inner wall. This temperature gradient causes heat conduction to the inner barrel. This phenomenon, referred to as thermal inertial, is a combined result of several factors such as the thermal conductivity of the metal barrel, the thermal mass of the barrel, the polymer material being processed, the location Received: Revised: Accepted: Published: 14759

April 21, 2012 October 18, 2012 October 19, 2012 October 20, 2012 dx.doi.org/10.1021/ie301036c | Ind. Eng. Chem. Res. 2012, 51, 14759−14770

Industrial & Engineering Chemistry Research

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of the thermocouple and the heater, and the power of the heater.7,8 It can directly influence the rising time of the temperature and the heating process. Also, barrel temperature dynamics exhibit significant time delays. Third, the extruder barrel heating source is divided into multiple zones, but they are all serving the same barrel. Unavoidably, there are strong interactions between adjacent heating zones caused by the temperature gradient through the barrel.9 These difficulties create tight requirements for precise modeling and sophisticated control strategy for barrel temperature. Previous studies focused on extrusion process control were limited. Some researchers attempted to use theoretical models to control melt pressure.10−12 The effect, however, was not that obvious. Although the models were in qualitative agreement, they lacked sufficient accuracy of quantitative correspondence because inadequate rheological theory was used. To stabilize the extrusion output flow rate, in industry the melt gear pump is adopted to eliminate unexpected surges and drifting output rates.13,14 Although the gear pump can precisely adjust the melt pressure, it is expensive and complicated to use. A traditional proportional integral derivative (PID) controller has been applied to control the melt pressure.15 It was well-tuned, but the pressure response is normally slow. An artificial neural network (ANN) scheme has also been designed and successfully implemented to pressure control.16 It was shown that the ANN was capable of predicting the pressure satisfactorily but needed a large amount of training data. A fuzzy control algorithm was adopted to decouple melt pressure and melt temperature control, but the precision of melt pressure control result was unsatisfactory.17 For barrel temperature control, conventional on−off and selftuning PID controllers are most widely used.15 These controllers provide good robustness, but at the cost of poor transient performance, usually with significant overshoots during the startup period and oscillations at steady state. On the other hand, model predictive control is more applicable for processes with slow dynamics and large dead-time.18 Multivariable dynamic matrix control has been proposed to control melt temperature by Dubay et al.,4,9 which proved to have good tracking performance and robustness, but the computational load was heavy because of the large number of step response coefficients involved in depicting the “integral” characteristic of the barrel temperature. In this work, a multivariable generalized predictive control (MGPC) is adopted to deal with the above problems, which has been shown to be effective for extrusion19,20 and injection molding control.21,22 It performs well in both the start-up period and steady state during operation for barrel temperature control. However, when the melt pressure is changed, significant temperature variations occur especially in the zones close to the hopper, due to the change of polymer feeding rate and the shear heating caused by rotating screw. Thus, feed-forward control will be incorporated into the MGPC to compensate for this sudden temperature variation. The overall control architecture proposed in this paper is constructed by two separate control loops: a single-input−singleoutput (SISO) generalized predictive control (GPC) control of melt pressure at die output and a multi-input−multioutput (MIMO) GPC control of barrel temperatures. GPC control strategy is simple and effective with high control performance. Compared with the traditional pressure control scheme it omits the use of gear pumps. Experiments show that the control system is robust against variations of operating conditions including different materials and set points. The effectiveness of the entire control system has been proved by product quality; in this paper, it is indicated by the dimensional stability.

2. DESIGN OF THE CONTROL SYSTEM The melt pressure mainly depends on the screw revolution speed, which is driven by the convertor’s commanding voltage. Similarly, the barrel temperatures mainly rely on the energy brought by the heater, although in practice it is well-known that the shear heat caused by screw revolution contributes to a significant percentage of the total heat generation.23 Screw revolution speed is a manipulated variable in the melt pressure control system, and in the barrel temperature control loop it can be viewed as a disturbance. It must be noted that the dynamics of barrel temperatures and pressure are significantly different:15for the extruder used in this paper, the time constant of the barrel temperature is about 40 min while the settling time of the pressure is about 5 s. The overall extruder control system can be illustrated as follows. First, a SISO GPC is designed to control the melt pressure. The control system is illustrated in Figure 1. Simultaneously, an

Figure 1. Pressure closed-loop control.

MIMO GPC is adopted to control the barrel temperatures. While melt pressure changes, a feed-forward controller is added to the GPC control system to suppress the disturbance. The entire temperature control system is schematically shown in Figure 2

Figure 2. Barrel temperatures closed-loop control.

3. MATHEMATICAL MODELING In GPC design, it is necessary to build a mathematical model with reasonable structure to predict future output of the controlled variable. There are mainly two ways to develop a suitable process model. One is based on the first principles; the other is using a system identification method. The building of the pressure dynamic model with the former way is a challenge because of the complexity, nonlinearity, and time-varying nature of the melt pressure process. The system identification method is utilized to identify pressure model order and parameters. The detailed design procedure can be found in 14760

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ref 24. The identified model is in the form of the controlled autoregressive integrated moving average (CARIMA) model. For barrel temperatures, on the other hand, the energy balance equation is easily constructed and can match the experimental finding. So in this research, the structure of the barrel temperature process model is obtained through theoretical analysis and calculation. On the basis of the deduced model structure, a recursive least-squares technique is then applied to identify model parameters, with better confidence and better understanding of the underlying physical dynamics. The derivation is given below. Figure 3 shows a physical model of the heating process. The heating barrel is divided into five zones in this paper, i = 1, 2, 3,

y(t) = [ y1(t) y2(t) y3(t) y4(t) y5(t)]T u(t) = [ u1(t) u 2(t) u3(t) u4(t) u5(t)]T ⎡ ε (t) ε (t) ε3(t) ε4 (t) ε5(t) ⎤T 2 ⎥ ε(t) = ⎢ 1 ⎣ m1C1 m 2C2 m3C3 m4C4 m5C5 ⎦ ⎡ α11 α12 0 0 0⎤ ⎥ ⎢ ⎢ α21 α22 α23 0 0 ⎥ ⎥ ⎢ A= ⎢ 0 α32 α33 α34 0 ⎥ B= ⎢ 0 0 α α α ⎥ 43 44 45 ⎥ ⎢ ⎢⎣ 0 0 0 α54 α55 ⎥⎦

⎡β ⎢ 1 ⎢0 ⎢ ⎢0 ⎢ ⎢0 ⎢ ⎢⎣ 0

0

0

0

β2 0

0

0

β3 0

0

0 β4

0

0

0

0⎤ ⎥ 0⎥ ⎥ 0⎥ ⎥ 0⎥ ⎥ β5 ⎥⎦

(3)

and α11 = − α21 =

α32 =

Figure 3. A schematic diagram of the extruder barrel.

4, 5. yi denotes the temperature of the ith barrel zone. mi represents the weight of each zone. Ci and ui denote the heat capacity and heating rate, respectively. riui(t) represents the applied thermal energy rate, Kij is the coefficient of thermal resistance between i and j adjacent zones. ε1(t) denotes the disturbance that may result from the energy lost to the environment or other sources of disturbance. The mathematical model of the barrel heating system with five heating zones is shown as follows: m1C1 m2C2

dy1(t ) dt dy2 (t ) dt

= r1u1(t ) + = r2u 2(t ) +

y2 (t ) − y1(t ) K 21 y3 (t ) − y2 (t ) K32

α43 =

α54 = β1 =

+ ε1(t ) +

y1(t ) − y2 (t )

dy3 (t ) dt

= r3u3(t ) +

y4 (t ) − y3 (t ) K43

+

y2 (t ) − y3 (t )

+

y3 (t ) − y4 (t )

m4 C4

dt

= r4u4(t ) +

K32

dy5 (t ) dt

= r5u5(t ) +

1 1 ⎛ 1 1 ⎞ + , α44 = − ⎜ ⎟, m4 C4 K43 m4 C4 ⎝ K54 K43 ⎠ 1 α45 = m4 C4 K54 1 1 , α55 = − m5C5K54 m5C5K54 r r r1 r2 , β2 = , β3 = 3 , β4 = 4 , m1C1 m 2 C2 m3C3 m4 C4 r5 β5 = m5C5

y(k) = Ay ̅ (k − 1) + Bu(k − d) + ε(k) y5 (t ) − y4 (t ) K54 y4 (t ) − y5 (t ) K54

⎡b1 0 ⎡ a11 a12 0 0 0 ⎤ ⎢ ⎢ ⎥ ⎢ 0 b2 ⎢ a 21 a 22 a 23 0 0 ⎥ ⎢ ⎢ ⎥ A̅ = ⎢ 0 a32 a33 a34 0 ⎥B = ⎢ 0 0 ⎢ ⎢0 0 a a a ⎥ 43 44 45 ⎢0 0 ⎢ ⎥ ⎢0 0 ⎢⎣ 0 0 0 a54 a55 ⎥⎦ ⎣

+ ε5(t ) (1)

Rearranging eq 1 into a matrix form gives the following: dy(t) = Ay(t) + Bu(t) + ε(t) dt

(5)

where

K43

+ ε4(t ) m5C5

1 1 ⎛ 1 1 ⎞ + , α33 = − ⎜ ⎟, m3C3K32 m3C3 ⎝ K43 K32 ⎠ 1 α34 = m3C3K43

Because the controller is actually a digital computer, a discrete time mathematical model must be developed. The discrete model can be obtained from the standard discretization procedure.25 The resulting discrete form barrel temperature model is shown in the following equation:

+ ε3(t ) dy4 (t )

1 1 ⎛ 1 1 ⎞ + , α22 = − ⎜ ⎟, m2C2K 21 m2C2 ⎝ K32 K 21 ⎠ 1 α23 = m2C2K32

(4)

K 21

+ ε2(t ) m 3C 3

1 1 , α12 = m1C1K 21 m1C1K 21

0 0 0⎤ ⎥ 0 0 0⎥ ⎥ b3 0 0 ⎥ ⎥ 0 b4 0 ⎥ 0 0 b5 ⎥⎦ (6)

(2)

ε(k) is a white Gaussian noise vector with zero mean, and d denotes the discrete-time time delay.

where 14761



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Figure 4. Illustration of generalized predictive control.

measurement noise. In this project, ff is set to 0.98, as this balanced value produces good estimates.

Because the transfer function model parameters cannot be easily calculated, it can be estimated through system identification. The recursive least-squares estimation method (RLSE) can be used here to identify the model parameters. It is constructed by means of the following equations:26

4. GPC DESIGN GPC is adopted in this paper for melt pressure and barrel temperature control because of good performance as well as robustness. The GPC algorithm was initially proposed by Clark et al.27,28 and has been widely utilized for chemical and petrochemical processes control from then on. The objective of GPC design is to obtain a sequence of future control signals through minimizing a predefined cost function, which is the expectation of a quadratic function measuring the distance between the predicted system output and the predefined reference sequence over the prediction horizon together with the penalty to the control input change. An illustration of a generalized predictive control is depicted in Figure 4. The SISO GPC for melt pressure control is described first in this section, followed by the extension to the MIMO form for the barrel temperature control. A brief introduction to SISO GPC is given below. With the shift operator z−1 and the identified system parameters introduced in the previous section, the system model can be written using a CARIMA form as shown below:

θi(k) = θi(k − 1) [p (k − 1)φi(k)(yi (k) − φiT (k)θi(k − 1))] + i [ff + φiT (k)pi (k − 1)φi(k)] pi (k) =

pi (k − 1) −

[pi (k − 1)φi(k)φiT (k)pi (k − 1)] [ff + φiT (k)pi (k − 1)φi(k)]

ff (7)

where i = 1, 2, 3, 4, 5 θ1(k) = [ a11(k) a12(k) b1(k)]T θ2(k) = [ a 21(k) a 22(k) a 23(k) b2(k)]T θ3(k) = [ a32(k) a33(k) a34(k) b3(k)]T θ4(k) = [ a43(k) a44(k) a45(k) b4(k)]T θ5(k) = [ a54(k) a55(k) b5(k)]T

A(z −1)y(k) = B(z −1)u(k − d) + C(z −1)ε(k)/Δ

φ1(k) = [ y1(k − 1) y2 (k − 1) u1(k − d)]T

(9)

where y(k) and u(k) are the controlled and manipulated variables, respectively. ε(k) is an uncorrelated sequence of random noise, and k represents the sampling interval. The noise is assumed to be white, i.e., evenly distributed in all frequency ranges and with a mean value equal to zero. d denotes process dead time. The operator Δ is defined as 1 − z−1,

φ2(k) = [ y1(k − 1) y2 (k − 1) y3 (k − 1) u 2(k − d)]T φ3(k) = [ y2 (k − 1) y3 (k − 1) y4 (k − 1) u3(k − d)]T φ4(k) = [ y3 (k − 1) y4 (k − 1) y5 (k − 1) u4(k − d)]T

A(z −1) = 1 + a1z −1 + a 2z −2 + ......anz −n

φ5(k) = [ y4 (k − 1) y5 (k − 1) u5(k − d)]T

B(z −1) = 1 + b1z −1 + b2z −2 + ......bmz −m

(8)

pi(k) is a weighting matrix, and ff is a forgetting factor which decides how fast the model is updated. The range for ff is 0 < ff ≤ 1; the smaller the ff, the faster the estimator can track the model variation but at the cost of more sensitive estimation to

C(z −1) = 1

(10)

The generalized predictive control law is derived by minimizing the following cost function: 14762



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f (k + d + j − 1) = [Gd + j − 1(z −1) − g0 − g1z −1 − ···

J(N1 , N2 , Nu , λ , Δu) N2

=

− z −(j − 1)gj − 1]z jΔu(k − 1) + Fj(z −1)y(k)

∑ [y ̂(k + j|k) − w(k + j)]2

(18)

j = N1

−1

(11)

j=1

ŷ = [ y ̂(k + d) y ̂(k + d + 1) ··· y ̂(k + d + N − 1)]T Δu = [Δu(k) Δu(k + 1) ··· Δu(k + N − 1)]T f = [ f (k + d) f (k + d + 1) ··· f (k + d + N − 1)]T ⎡ g 0 ⎢ 0 ⎢ g g0 G= ⎢ 1 ··· ⎢ ··· ⎢g g ⎣ N−1 N−2

w(k + n) = αw(k + n − 1) + (1 − α)Sp(k + n) (12)

Consequently, from the definition above and with: w = [ w(k + d) w(k + d + 1) ··· w(k + d + N − 1)]T

Equation 11 can be deduced as follows: J = (w − GΔu − f)T (w − GΔu − f) + Δu TλΔu

−1

y ̂(k + j|k) = Ej(z )B(z )Δu(k + j − d) + Fj(z )y(k) + Ej(z −1)ε(k + j)

(13)

Δu = (GTG + λ I)−1G(w − f)

where Ej(z ) and Fj(z ) are obtained by recursively solving the following Diophantine equation: −1

−j

−1

I = Ej(z )ΔA(z ) + z Fj(z )

(14)

u(k) = u(k − 1) + Δu(k)

Notice that because the degree of Ej(z ) is j − 1, the noise terms of eq 13 are all in the future. Therefore, the expectation of E[ε(k)] is equal to 0. The predicted value y(k + j) is given by: (15) −1

−1

where Ej(z )B(z ) = Gj(z ), the degree of Gj(z ) is less than j, and the prediction can now be further written as:29 y (̂ k + j|k) = Gj(z −1)Δu(k + j − d) + Fj(z −1)y(k)

(23)

The procedure is repeated at the next sample time. The important advantage of this receding horizon approach is that it can overcome model mismatch using the feedback information and suppress unmeasured disturbances. The entire deduction above is based on the SISO GPC algorithm. It can be implemented directly to melt pressure control. For barrel temperatures control, however, it will be extended to the MIMO system. For the MIMO system with five inputs and five outputs, in the CARIMA model formulation (eq 9), y(k), u(k), and ε(k) become 5 × 1 output vector, 5 × 1 input vector, and 5 × 1 noise vector at time k, respectively. A(z−1) is equal to 1 − A̅ (z−1) in the temperature model (eq 5). A(z−1), B(z−1) and C(z−1) are all 5 × 5 matrices. Then, following a similar derivation, the controller for MIMO can be readily obtained.

y ̂(k + j|k) = Ej(z −1)B(z −1)Δu(k + j − d) + Fj(z −1)y(k) −1

(22)

In GPC, only the first row of Δu, i.e., Δu(k), is entered into the process:

−1

−1

(21)

The minimum of J can be found by making its gradient equal to zero, The control signal u can thus be calculated as:

−1

−1

··· 0 ⎤ ⎥ ··· 0 ⎥ ⎥ ··· 0 ⎥ ··· g0 ⎥⎦ (20)

α is a tracking speed factor between 0 and 1; a value close to 1 leads to a slower but more robust control while a value close to 0 results in faster and more aggressive control. The GPC control law is obtained by minimizing this objective function (11). The j-step ahead output prediction can be written as follows: −1

(19)

where the vectors are all N × 1(N = N2 − N1 + 1)

w(k) = y(k)

n = 1......N2

+ ···. Then eq 16 can be

ŷ = GΔu + f

where ŷ(k + j/k), a j-step ahead prediction of the system output, is based on the identified model at current sampling time k. N1 and N2 are minimum and maximum prediction horizons. Nu is the control horizon. λ is a positively defined weighting matrix. w(k + j) is the future output reference sequence, and it is usually a smooth approximation from the current value of the output y(k) toward the known reference trajectory Sp(k + j) with the first-order system:

−1

−2

where Gj(z ) = g0 + g1z + g2z written in the vector form:

Nu

+ λ ∑ [Δu(k + j − 1)]2

−1

−1

(16)

Consider the following j-step ahead optimal predictions: y ̂(k + d|k) = Gd(z −1)Δu(k) + Fd(z −1)y(k) y ̂(k + d + 1|k) = Gd + 1(z −1)Δu(k + 1) + Fd + 1(z −1)y(k) ⋮

5. FEED-FORWARD CONTROL The MGPC controller can control the temperature well at normal operation. However, when operation status changes, e.g., during screw start, stop, or speed changes, the barrel temperatures may deviate from the set points due to the sudden disturbance. The heat required for the melting of polymer granules comes from two sources, the shear heating generated by screw rotation and the conductive heat from barrel heaters. Shear heating takes a significant percentage of the total heat generation; a

y ̂(k + d + j − 1|k) = Gd + j − 1(z −1)Δu(k + j − 1) + Fd + j − 1(z −1)y(k) (17)

Here ŷ(k + d + j − 1|k) consists of two terms: one depending on future control actions yet to be determined and the other corresponding to the free response of the process. Let f(k + d + j − 1) be that component of ŷ(k + d + j − 1|k) composed of signals which are known at time k, so 14763



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change in the screw rotary status, therefore, can cause a strong disturbance in the barrel temperature control. Thus, it can be treated as the dominant disturbance, while the variation of the environmental temperature, the measured disturbance, can be viewed as the other disturbance. The magnitude of the temperature disturbance is closely correlated to the variation of the melt pressure. It can be easily certified by a feedbackonly control result as shown in Figure 7. From this figure, it is obvious that at a certain pressure p1, the controlled input signal is stable around a certain steady-state value u1. After the melt pressure is changed to a new value p2, to maintain barrel temperature constantly, the heating rate is adjusted to a new value u2. On the basis of this observation, a steady-state relationship between melt pressure and heating rate is established. When melt pressure changes, a feed-forward signal uff is added to the temperature control system to compensate for this change; for example in the above case, when pressure changes from p1 to p2, uff = u2 − u1.

Figure 5. Structure of single screw extrusion machine.

The extrusion barrel is divided into five heating zones including a die zone, a screen pack zone, and three barrel zones, zone 1 to zone 3, aligned from the screen pack to the hopper. Each zone is equipped with an independent heater to heat and melt the polymer. The capacities of the heaters for the die, screen pack, and the other three zones are 400, 560, and 500 W, respectively. The temperatures are measured by K-type industrial grade thermocouples. They are located at about 5 mm away from the inner wall. In the following analysis, the temperature measurements from the heaters will be named die, screen, zone 1, zone 2, zone 3. Zero-crossing solid-state relay (SSR) was used to actuate the heater using pulse width modulation (PWM). All the measured variables are collected by a PC-104 computer via data acquisition board. The pressure and temperature signal acquisition is performed at 1000 Hz sampling rate, 10 Hz for analog voltage, 0.2 Hz for SSR, and 1 Hz for screw revolution speed. The control period for melt pressure and barrel temperatures are 100 ms and 15 s, respectively. The entire signal acquisition and control algorithms are implemented on the PC104 computer using C language under a real-time Linux-based operating system.

6. THE COMBINED CONTROLLER To achieve the benefits of feedback controller together with the feed-forward controller, the combination of feedback and feedforward controller can be used. The feed-forward control action compensates for the dominant disturbance while the feedback controller is responsible for handling other minor and random disturbances to maintain the controlled variable at the desired values. In this work, the ultimate controller can be formulated below: ⎧ u(k − 1) + Δu(k) ⎪ ⎪ for all the time excluding melt pressure changes u(k) = ⎨ ⎪ u(k − 1) + Δu(k) + uff ⎪ ⎩ at the time melt pressure changes

8. EXPERIMENTAL RESULTS AND DISCUSSION 8.1. Melt Pressure Control Results. To achieve a high product quality for the continuous extrusion process, melt pressure stability must be guaranteed. Failure to maintain pressure stability can lead to quality issues such as inconsistent profile and dimensional defect problems. In the extrusion process, the dynamic behavior of melt pressure is much faster compared with barrel temperatures; the settling time is only about 5 s. The melt pressure experimental test must evaluate the set point tracking performance as well as the robustness. Here tracking performance means how fast and how precise the controlled variable tracks the output, and it can be quantitatively represented by the response time and steady-state variation. Ideally, the steadystate error must remain as small as possible to ensure consistency of product quality, which is represented by the variation of the product quality. The following experiments were performed to observe whether this goal was achieved. The above proposed SISO GPC has been implemented and experimentally tested. The controller’s parameter is tuned to be conservative, the prediction horizon N1 is chosen as the process delay (one sample), N2 is about its process settling time plus process delay (ten + one samples), and control horizon Nu is selected as one sample to reduce the computation load. Weighting factor, λ, is chosen to be 5.0 to penalize input change. All the controller parameters are carefully chosen to ensure that the overall control system can meet the required performance specifications. The experiment is conducted with material PP and a rod die, and the temperature profiles of the heater from die to zone 3 are set to be 200 °C, 200 °C, 190 °C, 180 °C, and 170 °C, respectively.

(24)

where k is the sampling time and uff is the feed-forward action. The entire control system can be limned by Figure 2. In GPC strategy, the measurements and the control movements Δu(k) are calculated in eq 22. The combined formulation (eq 24) does not affect the GPC calculation at all. With the above feed forward control compensator, the dominant disturbance can be compensated.

7. EXPERIMENTAL SETUP In this research, all the experiments for process identification and control implementation were conducted on a HIGHRICHJA single screw extruder (model HJ-25). Figure 5 shows a schematic diagram of the extruder machine. The pressure transducer is placed on the die after the screen pack, which is mounted at the contact interface between metal and the material. The geometry of the die is rod as shown in Figure 6. In the melt pressure control system, the controlled variable is melt pressure, while the manipulated variable is analog voltage which manipulates the converter’s frequency and then determines the screw revolution speed. The analog voltage can be modulated between 0 V and 1 V corresponding to a screw revolution speed between 0 rpm and about 60 rpm. The screw revolution speed is measured by an incremental encoder with maximum speed range of 10 000 rpm and a resolution of 1000 pulses per revolution. 14764



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Figure 6. Geometry of rod die.

Figure 9. Pressure control result (PID): 48 bar.

Figure 7. Heating rate vs melt pressure (to maintain constant temperature).

variation for the proposed SISO GPC is about ±0.1 bar, which is smaller than the PID controlled result of ±0.2 bar. The comparison clearly demonstrates the superior performance of the proposed SISO GPC controller against the PID control. Case 8.1.2. Comparison of Step-Change Control Result between SISO GPC and PID. A set point step change from 48 to 53 bar is applied to further test the pressure controller. The result plotted in Figure 10 clearly shows that the response curve almost

Case 8.1.1. Comparison of Steady-State Pressure Control Result between SISO GPC and PID. The first set of experiments is performed to test the set-point tracking performance in the machine steady state. The proposed SISO GPC control result is compared with a commercial PID controller result, as shown in Figure 8 and Figure 9, respectively, using a constant pressure set

Figure 8. Pressure control result (GPC): 48 bar. Figure 10. Pressure step control result (SISO GPC): 48 to 53 bar.

point (48 bar) and PP as the processing material. The temperature set points from die to zone 3 are 200 °C, 200 °C, 190 °C, 180 °C, and 170 °C, respectively. It can be seen that the steady-state

overlaps with the set point, proving the excellent performance of the SISO GPC pressure controller. Figure 11 gives the step-change 14765



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Figure 12. MIMO GPC control results of the barrel temperature in the idle state.

Figure 11. Pressure step control result (PID): 48 to 53 bar.

response of the PID controller for comparison. From these figures, it is obvious that the proposed SISO GPC controller takes only 5 s to follow the set point step change without obvious overshoot while the time for PID controller is about 10 s. This result confirms the advantage of GPC in tracking performance. 8.2. Barrel Temperature Control Results. The melt pressure control starts only when the machine is operating. The barrel temperature, however, needs to be controlled from the time that the machine is powered on. An ideal barrel temperature controller should bring the barrel temperatures to set points during startup as soon as possible, while avoiding large overshoots, and trace the desired set points profile tightly during the machine idle state. When the machine is in the operation stage, it should be able to reject the disturbance exerted to the system from different sources. Therefore, the barrel temperature controller must meet the following requirements: (1) in the start-up stage, all the zone temperatures must reach the set points quickly and with overshoots less than 1%; (2) in idle and operating stages, all heating zones shall be controlled with a small steady-state variation; (3) when the machine is switched from idle stage to operation stage, the temperature variation shall be minimized. The following design and experiments were conducted to verify whether all these goals were achieved. During the entire machine operation stage, the SISO GPC pressure controller is also in use. The controller’s parameters must be properly tuned first to achieve these objectives. Controller horizon N1 is chosen as the process delay (5 samples), and N2 is the closed-loop process settling time plus process delay (25 + 5 samples), which is shorter than the open-loop time constant (nearly about 150 samples) because of the integrating characteristic of the barrel temperatures.30 Control horizon Nu is selected as 1 sample to reduce the computation load. Weighting factor λ is chosen to be 7.0 to penalize input change. Case 8.2.1. MIMO GPC Control Results of Barrel Temperature in the Idle Stage. The first set of experiments was performed to test the set-point tracking performance in the machine idle state using the proposed MIMO GPC controller. The experiment was conducted over 6000 samples with a sample interval of 1 s, i.e., 100 min. Figure 12 plots the control result with constant temperature set points 200 °C, 200 °C, 190 °C, 180 °C, and 170 °C, respectively. The processing material was polypropylene (PP). It is clearly shown that the temperatures can be controlled from the ambient temperature to the set points quickly and smoothly, the rising time is about 18 min, and the maximum overshoot is only about 1 °C. Case 8.2.2. Comparison of the Temperature Control Results between MIMO GPC and PID. A well-tuned commercial PID

controller is tested on the same extruder. Because die temperature is closest to melt temperature, this zone’s temperature is selected for comparison purposes. Figure 13 compares a typical

Figure 13. Comparison of die temperature control result between MIMO GPC and PID.

control result of the proposed controller, as plotted by a black line, and a control result of the commercial PID controller, as plotted by a gray line. Start-up time is about 18 min for the GPC and 34 min for the PID controller, and overshoots are less than 1 °C and 10 °C for the GPC and PID, respectively. Moreover, the steady-state variation for GPC is about ±0.15 °C, while for PID it is about ±0.5 °C. This comparison demonstrates the superior performance of the proposed GPC controller against conventional PID control. Case 8.2.3. GPC Control Results of Barrel Temperatures in the Operation Stage. During extrusion, when the screw starts to rotate, it generates significant shear heat to the melting system. Therefore, when the extruder switches from idle stage to normal operation stage, the temperature control system undergoes a large disturbance. The following experiment was performed to test the load rejection capabilities of the proposed methodology. To fully test the control performance, the following group of experiments was conducted for about 1 h. Figure 14 shows the barrel temperature control results of the proposed GPC controller. Table 1 lists each zone’s temperature variation and mean value together 14766



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Figure 15. GPC control results of die temperature in the operation state.

Figure 14. GPC control results of barrel temperatures in the operation state.

with the temperature variations controlled by the commercial PID controller. It is obvious that all the temperatures are controlled within a narrow range, less than 0.5 °C by the GPC controller. The relatively big variation of the last zone is due to the cool water in the cooling ring. These control results verified the effectiveness of the proposed control strategy. Case 8.2.4. GPC and Commercial PID Control Result of Die Temperature in the Operation Stage. This control result is again compared with the commercial PID controller’s control results. Figure 15 shows the control results of the proposed GPC controller and the commercial PID under the pressure of 48 bar. It is shown in this figure that for the proposed GPC controller, when the machine is in normal operation, the temperature is maintained at the set point with a small variation of ±0.15 °C. However, for the commercial PID controller, there is a big drop in temperature, due to the temperature dynamics change from idle to machine operation; the temperature variation is more than 3 °C. Case 8.2.5. MIMO GPC during the Start-up Period of the Operation Stage with PP. With material PP, a melt pressure change was introduced intentionally as follows: the extruder operates at a melt pressure of 60 bar until the process reaches the steady state, and then the melt pressure changes to 80 bar first. A series of step changes on the melt pressure set point is applied similarly, as plotted by the dotted line. In this experiment, there is no feed-forward information for the die pressure change. The barrel temperature of zone 3 is selected for demonstration purposes and plotted as a solid line in Figure 16. The temperature under control is clearly affected by the melt pressure change, and the disturbance is significant, with a variation larger than 4 °C. Case 8.2.6. MIMO GPC with Feed-Forward Control during the Start-up Period of the Operation Stage with PP. The feed-forward controller is then incorporated into the MIMO GPC and tested with the same setting of die pressure step changes. The results are shown in Figure 17. It is obvious that the control performance is significantly improved from the feed-forward action, and the temperature variation is reduced to

Figure 16. Performance of zone 3 without feed-forward control.

Figure 17. Performance of zone 3 with feed-forward control.

about 2 °C. A similar approach has been applied to other barrel zones with satisfactory control results. Case 8.2.7. Barrel Temperature Control Result with Material LDPE. A different material, low-density polyethylene (LDPE), was used to test the robustness of the proposed controller. The temperature set points of the five zones were changed to 180 °C,

Table 1. Each Zone’s Temperature Variation and Temperature Mean controller

temperature

die

screen

zone 1

zone 2

zone 3

GPC

mean variation variation

200.00 ±0.15 ±1.75

200.01 ±0.15 ±1.64

190.02 ±0.07 ±1.66

180.00 ±0.07 ±2.08

169.99 ±0.40 ±2.79

PID

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Figure 21. Histogram comparison of the GPC control system and commercial PID.

Figure 18. Barrel temperature control results with material LDPE.

good indication of process stability. Variations of the process variables can immediately cause drifts in the product diameter.31 Case 8.3.1. Comparison of the Commercial Controller’s Control Results with Proposed Controller’s Control Results (material: PP melt pressure set point: 53 bar). As introduced previously, the process variables such as barrel temperatures and melt pressure have a close relationship with product quality. Precise control of these process variables have been accomplished in the previous sections. It is desirable to verify the effectiveness of the developed process control systems with the product quality. The first group of experiments was conducted under a melt pressure of 53 bar with material PP, the puller’s speed was set to be 4.1 rpm, the product diameter was sampled every 17 s, and 200 samples were taken for each test condition. Figure 19 shows the product diameter produced using the proposed GPC controller. The standard deviation of the product diameter was calculated to be 0.05041. Figure 20 shows the product diameter produced by the commercial PID controller for comparison purposes, and the standard deviation of the diameter was 0.169. From Figures 14 and 15, it is obvious that by using the proposed GPC control system, the product quality consistency represented by the diameter has been improved significantly; the standard deviation has been reduced about 70% compared to that of the commercial control system. Figure 21 compares the histograms of these two cases to give a better understanding of the product quality improvement. The dashed line shows the diameter distribution with the commercial controller while the solid line plots the one with the proposed GPC control. It is obvious that the latter gives a much better quality consistency with narrower distribution. Case 8.3.2. Comparison of Commercial Controller’s Control Results with Proposed Controller’s Control Results (material: LDPE melt pressure set point: 41 bar). To further test the robustness of the proposed control system, a second group of experiments was performed with a different material, low-density polyethylene (LDPE), whose property is significantly different from that of PP. The melt pressure and puller speed were controlled at 41 bar and 4.1 rpm, respectively. The product diameter was sampled every 17 s again, and 200 samples were taken for each experiment. Figure 22 shows the product diameter using the proposed GPC control, while Figure 23 shows that produced by the commercial PID controller. These two figures give the conclusion that the product consistency of the proposed GPC control can be improved from 0.0521 to 0.0142, about 73%

Figure 19. Product diameter produced by the proposed controller.

Figure 20. Product diameter produced by the commercial controller.

175 °C, 170 °C, 165 °C, and 160 °C, respectively, to meet the processing requirements. Figure 18 shows the control result using this new material. It clearly proves the effectiveness of the GPC and its robustness against different processing materials. 8.3. Product Quality. In all the previous experiments, a die with rod geometry was used. The product diameter is selected as the product quality measurement in the quality analysis for the following reasons. First, the product diameter has a close relationship with the other quality properties, particularly the dimensional properties. For example, it has a strong linear relationship with product weight. Second, product diameter is a 14768



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to control extrusion melt pressure and barrel temperatures, respectively. For melt pressure, the control results show that it traces the set point precisely. With proper tuning, the extruder barrel temperatures can achieve a fast start-up heating without significant overshoot and showed high precision during steadystate operation. For the barrel temperature, a feed-forward controller has been incorporated into the MGPC to compensate for the disturbance caused by the change in melt pressure changes. The effectiveness of the proposed control system was verified quantitatively using the product diameter as the quality measurement. The experimental results demonstrated the effectiveness, robustness, and significant improvements of the proposed control system compared to the commercially available PID control.

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ACKNOWLEDGMENTS The authors acknowledge the financial support from the Fundamental Research Funds for the Central Universities 2012FZA5010, Guangzhou scientific and technological project (2012J5100032) and Nansha district independent innovation project(201103003).

Figure 22. Product diameter produced by the proposed controller with material LDPE.

■

AUTHOR INFORMATION

Corresponding Author

*Tel.: +86-571-87951011; e-mail: [email protected]. Notes

The authors declare no competing financial interest.

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REFERENCES

(1) Chung, C. I. Extrusion of Polymers: Theory & Practice; Hanser Gardner Publications: Munich, 2000. (2) Rauwendaal, C. Polymer Extrusion, 4th ed.; Hanser: Munich, 2001. (3) Patterson, T. D. I. The Dynamic Behaviour of Extruders. Annu. Tech. Conf. - Soc. Plast. Eng. 1978, 24, 483. (4) Dubay, R. Self-Optimizing MPC of Melt Temperature in Injection Moulding. ISA Trans. 2002, 41, 81. (5) Tadmor, Z.; Lipshitz, S. D.; Lavie, R. Dynamic Model of a Plasticating Extruder. Polym. Eng. Sci. 1974, 14, 112. (6) Pfeiffer, B.-M. Towards ‘Plug and Control’: Self-Tuning Temperature Controller for PLC. Int. J. Adapt. Control Signal Process. 2000, 14, 519. (7) Yao, K.; Gao, F. R. Optimal Start-up Control of Injection Molding Barrel Temperature. Polym. Eng. Sci. 2007, 47, 254. (8) Berins, M. L. SPI Plastics Engineering Handbook of the Society of the Plastics Industy Inc., 5th ed.; Springer-Verlag: London, 1991. (9) Rickey, D.; Adam, Bell C.; Yash; Gupta, P. Control of Plastic Melt Temperature: A Multiple Input Multiple Output Model Predictive Approach. Polym. Eng. Sci. 1997, 37, 1550. (10) Horrobin, D. J.; Nedderman, R. M. Die Entry Pressure Drops in Paste Extrusion. Chem. Eng. Sci. 1998, 53, 3215. (11) Lang, W. M. U. Development of a Mathematical Model for the Calculation of the Pressure Drop in Extrusion Dies. J. Reinf. Plast. Compos. 1998, 17, 1110. (12) Abeykoon, C.; Kang, L.; Martin, P. J.; Kelly, A. L.Modelling of Melt Pressure Development in Polymer Extrusion: Effects of Process Settings and Screw Geometry. Proceedings of 2011 International Conference on Modelling, Identification and Control (ICMIC), June 26−29, 2011; p 197. (13) McKelvey, M. J.; Lavin, S. Gear-Pump-Assisted Plasticating Extrusion-Par II: Applications. Adv. Polym. Technol. 1982, 2, 171. (14) Malloy, R. A. Closed Loop Inlet Pressure Control for Melt Pump Extrusion. Annu. Tech. Conf. - Soc. Plast. Eng. 1984, 604. (15) Previdi, F.; Savaresi, S. M.; Panarotto, A. Design of a Feedback Control System for Real-Time Control of Flow in a Single-Screw Extruder. Control Eng. Practice 2006, 14, 1111.

Figure 23. Product diameter produced by the commercial controller with material LDPE.

Figure 24. Histogram comparison of the GPC control system and commercial PID.

in terms of standard deviation of the diameter. Figure 24 compares the histograms of these two cases. Again the dashed line shows the product diameters with the commercial PID control, and the solid line plots those with the proposed GPC control. The results further verify the robustness and effectiveness of the proposed controller.

9. CONCLUSION In this work, two separate control loops, a SISO GPC controller and an MIMO GPC controller, were designed and implemented 14769



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(16) Li, Y. Y.; Bridgwater, J. Prediction of Extrusion Pressure Using an Artificial Neural Network. Powder Technol. 2000, 108, 65. (17) Wen, S.; Qu, J. Fuzzy Decoupling Control of Melt Temperature and Melt Pressure for Plastic Electromagnetism Dynamic Extruder; IET Conference Publications: Hangzhou, 2006; p 1893. (18) Yao, K.; Gao, F. R.; Allgower, F. Barrel Temperature Control During Operation Transition in Injection Molding. Control Eng. Practice 2008, 16, 1259. (19) Lu, C. H.; Sai, C. C. Multivariable Self-Tuning Temperature Control for Plastic Injection Molding Process. Industrial Automation and Control: Emerging Technologies. Int. IEEE/IAS Conf. 1995, 702. (20) Tsai, C. C.; Lu, C. H. Multivariable Self-Tuning Temperature Control for Plastic Injection Molding Process. IEEE Trans. Ind. Appl. 1998, 34, 310. (21) Yao, K.; Gao, F. R. Adaptive GPC Control of Barrel Temperatures in Injection Molding. Annu. Tech. Conf. - Soc. Plast. Eng. 2005, 2, 813. (22) Huang, S. N.; Tan, K. K.; Lee, T. H. Adaptive GPC Control of Melt Temperature in Injection Molding. ISA Trans. 1999, 38, 361. (23) Rasid, R.; Wood, A. K. Effect of Process Variables on Melt Temperature Profiles in Extrusion Process Using Single Screw Plastics Extruder. Plast., Rubber Compos. 2003, 32, 187. (24) Ljung, L. System Identification: Theory for the User, 2nd ed.; Prentice-Hall: Upper Saddle River, NJ, 1999. (25) Seborg, D. E.; Edgar, T. F.; Mellichamp, D. A. Process Dynamics and Control, 3rd ed.; Wiley: New York, 2010. (26) Astrom, K. J.; Wittenmark, B. Adaptive Control, 2nd ed.; Addison-Wesley: New York, 1994. (27) Clarke, D. W.; Mohtadi, C.; Tuffs, P. S. Generalized Predictive Control–Part II Extensions and Interpretations. Automatica 1987, 23, 149. (28) Clarke, D. W.; Mohtadi, C.; Tuffs, P. S. Generalized Predictive Control–Part I. The Basic Algorithm. Automatica 1987, 23, 137. (29) Camacho, E. F.; Bordons, A. C. Model Predictive Control, 2nd ed.; Springer-Verlag: London, 2004. (30) Dougherty, D.; Cooper, D. J. Tuning Guidelines of a Dynamic Matrix Controller for Integrating (Non-Self-Regulating) Processes. Ind. Eng. Chem. Res. 2003, 42, 1739. (31) Harris, H. E. Extrusion Control Machine-Process-Product; Hanser: Munich, 2004.

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