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Two-Time-Dimensional Model Predictive Control of Weld Line Positioning in Bi-Injection Molding Zhixing Cao, Yi Yang, Jingyi Lu, and Furong Gao* Department of Chemical and Biomolecular Engineering, Hong Kong University of Science & Technology, Clear Water Bay, Kowloon, Hong Kong ABSTRACT: The application of bi-injection molding, a simple variation of the conventional injection molding process, is limited because of the lack of a good control of the weld line location. The weld line is an inevitable consequence of the meeting of two polymer melt fronts from different gates in the mold cavity. Positioning the weld line to avoid critical areas for the sake of aesthetics or mechanical strength is the key to the success of bi-injection molding. This remains a challenge for control engineers. This paper proposes a two-time-dimensional model predictive control for the process by combining a conventional model predictive control with an iterative learning control to explore the repetitiveness of the process. To predict the final position of the weld line, the approach takes the geometric dimension of the mold cavity into consideration. Simulation shows that the proposed method has good ability to control the weld line position ultimately within a narrow range of the desired position.

1. INTRODUCTION Plastics play an important role in modern society. More than 33% of all plastics are processed by injection molding, a typical processing method for transforming plastic granules into various plastic products of complex shapes with high productivity and versatility.1 Increasing requirements on end-product qualities may not be met by the conventional injection molding. As a result, new variations of injection molding process have been developed to extend the applicability, capability, and flexibility of the process to a higher level. The application of multicomponent injection molding among these emerging technologies can be found in many areas of plastics manufacturing, e.g. headlight reflectors, garden furniture, keyboard keys for computers, telephones, and power tools. Bi-injection molding is the simplest variation of the conventional injection molding in terms of process and machinery. A bi-injection molding machine typically consists of four major units: two injection units and hydraulic units, a clamping unit, and a control unit, as illustrated in a simplified schematics of Figure 1. The process can be roughly divided into three phases: (1) filling, (2) packing and holding, and (3) cooling. The process starts with the mold closed just before filling. In filling, both injection screws move forward simultaneously pushing two different polymer flow melts separately into the mold cavity through their runners and gates. During this period, the injection velocity is the control variable to ensure uniformity; the two flows meet in the cavity, resulting in a weld line where they meet. The location of weld line is determined by the advancement of these two polymer melt flow fronts. Also, around the same time, the cavity is nearly filled, leading to a rapid rising of the pressure in the mold cavity. This triggers a process called V/P transfer to switch the process from filling to packing and holding phase. In the packing and holding phase, some extra melt is “packed” into the mold cavity to compensate for the material shrinkage associated with cooling and solidification. Correspondingly, the controlled variable should be switched from injection velocity to packing pressure. Packing and holding phase lasts until the gates freeze off followed by cooling phase, during which the melts in © 2015 American Chemical Society

the cavity keep cooling until they are solidified enough to be ejected. Concurrent to cooling of the materials in the mold, polymer granules in the barrel are being melted and conveyed by the rotation of screw. Polymer melt accumulated in the front of the barrel generates a pressure to push the screw to retract to prepare for next batch. Finally, the mold opens, the product is ejected, and the process is ready for the next batch. The process procedure is depicted in Figure 2. Compared to other multicomponent injection molding, biinjection molding has the advantage of melt materials being injected simultaneously into the cavity via different gates, whereas in other techniques, the polymer melts are injected into the cavity in a sequential fashion leading to loss of productivity. Nevertheless, the appearance of weld line is inevitable for bi-injection molding. The mechanical property at the weld line location is lower than that at other locations, and the weld line degenerates the surface smoothness as well. Therefore, it is desirable to position the weld line in a noncritical area to avoid the above-mentioned issues. The lack of good weld line position controls hinders the application of bi-injection molding.2 The control of the weld line position to the desired location with high precision is an interesting and challenging problem for control engineers in the plastic processing industries. According to the best knowledge of the authors, there are very limited publications on such topics. Liu et al. employed the Taguchi method to optimize setting of process parameters (injection velocity, packing pressure, etc.) to achieve experimentally a maximum weld line strength in conventional injection molded parts;3 Wu and Liang conducted further research on the factors associated with weld line strength based on the Taguchi method by considering not only the process parameters but also geometric factors.4 These two studies Received: Revised: Accepted: Published: 4795

October 20, 2014 March 25, 2015 April 7, 2015 April 7, 2015 DOI: 10.1021/ie5041397 Ind. Eng. Chem. Res. 2015, 54, 4795−4804

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Industrial & Engineering Chemistry Research

Figure 1. A simplified schematic diagram of bi-injection molding machine.

2. PROBLEM DESCRIPTION The location of the weld line is determined by the two melt front advances from their respective gates. The advancement distance is essentially an integration of injection velocity. At the same time, it is essential to control the velocity properly in the filling phase, as suggested by Wu et al., because it has a large impact on weld line strength.4 This paper focuses on the development of 2DMPC control of injection velocity to achieve the following two simultaneous objectives for bi-injection molding: • Positioning of the weld line to avoid structurally or aesthetically sensitive areas; • Tracking of injection velocity to a given profile.

focused on a single-gate scenario. As for the multigate cases, Zhai et al. designed an optimal set of gate locations to drive the weld line into the most desired area with the help of computer-aided design (CAD) and optimized the runner design to achieve an appropriate shape of weld line according to material balance.5 All these results are attributed to the mold or gate design, not an online control strategy. Chen et al. applied fuzzy control to position of weld line; however, they did not take multigate cases into account, and they did not explore the repetitive nature of injection molding processes.6 This paper intends to resolve the weld line position control problem of bi-injection molding based on two-time-dimensional model predictive control (2DMPC), which was first introduced by Shi et al.7 2DMPC is an appropriate way to improve control performance batch-to-batch via exploring the repetitiveness of batch processes, which has made it attractive for bi-injection molding control. Details about 2DMPC can be found in the review paper of Wang et al.8−12 The philosophy of 2DMPC combines conventional model predictive control (MPC), a typical method of feedback control with great success,13−19 with iterative learning control (ILC),20−24 a feedforward strategy. The contributions of this paper are as follows: First, a 2DMPC strategy is exploited to improve the precision of weld line positioning. Second, a practical method is proposed to predict the cavity cross-sectional area to avoid heavy computational load of the corresponding optimization problem. The paper is organized as follows: The Introduction provides background information on bi-injection molding and its associated weld line problem. Problem Description formulates the control problem of bi-injection molding weld line positioning. The control law is given and derived in Controller Design. The simulation is given in Numerical Simulation to demonstrate the performance of the proposed control strategy. Conclusions are drawn in the last section.

3. CONTROLLER DESIGN 3.1. Establishing the Model. The following set of equations describes the dynamics of injection molding in the filling phase.25 β ⎛ dPh dZ ⎞ = h ⎜qh − Ah ⎟ ⎝ dt Vh dt ⎠

(1a)

dvz 1 = (PhAh − PnA n − fv ) dt M

(1b)

dZ = vz dt

(1c)

βp ⎛ dZ ⎞ dPn ⎜A = − q p⎟ n ⎠ dt Vn ⎝ dt

(1d)

dVp dt

= qp

(1e)

Here, P, q, and A are the pressure, flow rate, and cross-sectional area, respectively. The subscripts “h” and “n” stand for the 4796

DOI: 10.1021/ie5041397 Ind. Eng. Chem. Res. 2015, 54, 4795−4804

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Figure 2. Workflow of bi-injection molding process.

hydraulic system and nozzle, respectively. Vh, Vn are respectively the volume in the injection cylinder and the nozzle. βh, βp are respectively the bulk modulus of the hydraulic oil and the polymer melt. M is the ram mass, and Z is the screw displacement

with vz being the ram velocity. f v is the friction force against the ram-screw movement. Vp is the instantaneous volume of the cavity filled by the polymer, qp being the flow rate of the polymer melt. 4797


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Industrial & Engineering Chemistry Research yk , i (t )Ab = Akf , i (t )Mkf , i(t )

It has been shown in the work of Rafizadeh and colleagues that it is quite accurate to approximate βh/Vh and βp/Vn as constant.25 Moreover, despite the nonlinearity of Pn and f v, they can be regarded as exogenous disturbances, and ILC is quite good at learning such kinds of unknown nonlinearities. Moreover, model predictive control can also attenuate the impact of these factors, and in the simulation later, both factors are considered as the worst casewhite noise. Additionally, the valves in this case are assumed to be servo valves, whose map between valve opening and flow rate is linear. Thus, it is sufficient to model the dynamics as linear as experimentally proven in the works of Gao and Yang.26−29 Without loss of generality, the dynamics for two injection units (with injection velocity as the output and valve opening as the input) is assumed as follows.

Figure 3. Illustration of the relation between injection velocity and melt front velocity.

where Ab is the cross-sectional area of the barrel, which is assumed to be a constant. Afk,i(t) is the cross-sectional area of cavity normal to the direction of melt flow front associated with the ith injection unit in the tth time instant and kth batch, and the superscript f means “front”. Mfk,i(t) is the melt front velocity associated with ith injection unit of the tth time instant and kth batch. From the equation above, one can see that the melt front velocity, Mfk,i(t), is decided explicitly if Afk,i(t) is known. The determination of Afk,i(t) will be discussed in a following section. The location of the weld line can be depicted with the central line being the reference. Then, the distance between the central line and the ultimate weld line position is defined as follows. The relationship is shown in Figure 4.

⎧ xk ,1(t + 1) = A1xk ,1(t ) + B1uk ,1(t ) ⎪ Injection unit 1: ⎨ ⎪ yk ,1(t ) = C1xk ,1(t ) ⎩ (2)

⎧ xk ,2(t + 1) = A 2 xk ,2(t ) + B2 uk ,2(t ) ⎪ Injection unit 2: ⎨ ⎪ yk ,2 (t ) = C2xk ,2(t ) ⎩ (3)

where u is the control input (valve opening), x, y for states and the output (injection velocity) respectively. A1 , A 2 ∈ n; B1 , B2 ∈ n ; and C1 , C2 ∈ 1 × nare coefficient matrices for the state space model. The first subscript k∈Z + stands for the batch index, the second subscript (1 or 2) for injection unit index. t∈Z + represents the time instant. Actually, every component of the state vector is approximately a linear combination of the two physical variables Pn and vz. This is followed by decomposing the system of eqs 2 and 3 into two subsystems, one evolving along the time direction, the other along the batch direction, by an operation named two-timedimensionalization, a common step in all 2DMPC algorithms.7 Timewise system:

Pf =

⎡ δxk ,1(t + 1)⎤ ⎡ A ⎤⎡ δxk ,1(t )⎤ ⎡ B1 ⎤⎡ rk ,1(t )⎤ ⎢ ⎥ ⎥=⎢ 1 ⎥+⎢ ⎥⎢ ⎥⎢ ⎢⎣ δx (t + 2)⎥⎦ ⎣ ⎢⎣ δx (t )⎥⎦ ⎣ ⎢⎣ r (t )⎥⎦ A B ⎦ ⎦ 2 2 k ,2 k ,2 k ,2

∫0

s*

[Mkf ,1(n) − Mkf ,2(n)] dn

(7)

Figure 4. Definition of Pf.

where s* is the solution of the integration equation

(4)

Batchwise system:

L=

⎡ ek ,1(t + 1)⎤ ⎡ ek − 1,1(t + 1)⎤ ⎡C A ⎤⎡ δxk ,1(t )⎤ ⎢ ⎥=⎢ ⎥ ⎥−⎢ 1 1 ⎥⎢ ⎢⎣ e (t + 1)⎥⎦ ⎢⎣ e ⎢⎣ δx (t )⎥⎦ ⎥⎦ ⎣ C A ⎦ + ( t 1) 2 2 k ,2 k − 1,2 k ,2 ⎤⎡ crk ,1(t )⎤ ⎡C1B1 ⎥ ⎥⎢ ⎢ − C2B2 ⎦⎢⎣ rk ,2(t ) ⎥⎦ ⎣

(6)

∫0

s*

[Mkf ,1(n) + Mkf ,2(n)] dn

(8)

where L is the total cavity length between two gates. Equation 8 implies that there should be a tactic included in the control algorithm to deal with an uneven length issue, which will be addressed later. An index indicating the instantaneous weld line position is introduced as follows.

(5)

where rk,i(t)≜ uk,i(t) − uk−1,i(t), i = 1,2 is the control input augment between two consecutive batches. δ is the batchwise backward differential operator. ek,i(t)≜ yd,i(t) − yk,i(t), i = 1,2 represents the injection velocity tracking error for the tth time instant, kth batch, and ith injection unit, with yd,i(t) being the desired velocity profile for the ith injection unit. The weld line position is determined by the velocity integration of two melt fronts. The material balance shows that the injection velocity and melt front velocity are governed by the following equation, also shown in Figure 3.

Pk(t ) =

∫0

t

[Mkf ,1(n) − Mkf ,2(n)] dn

(9)

The above equation is in continuous form, making it difficult to implement by a digital controller. Thus, it is necessary to discretize the above equation. When eq 9 is combined with eq 6, one can obtain t ⎛ y (i ) yk ,2 (i) ⎞ k ,1 ⎟⎟ Pk(t ) ≈ AbΔ ∑ ⎜⎜ f − f Ak ,2 (i) ⎠ i = 1 ⎝ Ak ,1(i)

4798

(10)


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geometry, irrespective of thickness variation along the flow length, but is valid only for the cavities that can be abstracted as a tube. More specifically, the cross-sectional area taken along the flow direction does not allow the existence of hole or discontinuity within its interior. The relation between Afk,i(t) and ϕi can be written down explicitly.

where Δ is the sampling period. This approximation can be arbitrarily accurate if Δ is sufficiently small. Generally speaking, if the cavity geometry is quite smooth and the injection velocity is quite low, then the performance will not deteriorate much for a little larger sampling period. To facilitate the controller design, a trajectory for desired position tracking is introduced, which can be calculated from desired injection velocity profile. ⎛ y (i ) yd,2 (i) ⎞ d,1 ⎟⎟ Pd(t ) = AbΔ ∑ ⎜⎜ f − f ( ) ( ) A i A i ⎝ ⎠ d,1 d,2 i=1

⎛ t − 1 y ( j) ⎞ k ,i ⎟, Akf , i (t ) = ϕi⎜⎜AbΔ ∑ f ⎟ A k , i ( j) ⎠ ⎝ j=1

t

(11)

i = 1, 2 (15)

f with Ak,i (1) =ϕi(0). Figure 5 illustrates the process of discretization of eq 15.

Therefore, with the position index and trajectory defined, we are ready to define a cost function for 2DMPC control. N

Jk (t ) =

∑[

ek̂ ,1(t + i|t )

2 Q

+ ek̂ ,2(t + i|t )

2 Q

]

i=1 N

+

∑ ∥Pk̂ (t + i|t ) − Pd(t + i)∥2R i=1 M−1

+

∑[

rk̂ ,1(t + i|t )

2 S

+ rk̂ ,2(t + i|t ) S2 ]

i=0

(12)

where Q, R, and S are positive weights; M and N are positive integers as prediction horizon and control horizon, respectively. Generally, N ≥ M. êk,j(t + i|t) and r̂k,j(t + i|t) stand for the injection velocity prediction error and predicted control augments between two consecutive batches, respectively, associated with the jth injection unit in the (t + i)th time instant and kth batch based on all the information before or at the tth time instant. P̂ k(t + i|t) is the predicted weld line position for the (t + i)th time and kth batch instant based on all the information before or at the tth time instant, which is governed by eq 10. The cost function defined in eq 12 considers the velocitytracking error as well as the position-tracking error. From the aspect of degree of freedom analysis, there are two variables that can be manipulated, e.g., rk,1(•) and rk,2(•), but there are three control objectives to be achieved, e.g., perfect velocity tracking for both injection units and perfect position tracking. However, the position trajectory is calculated from velocity trajectories as mentioned before, which implies that perfect position tracking is consistent with velocity tracking, or these three targets are able to be achieved simultaneously. By now, the remaining problem is how to carry out the prediction of P̂k(t + i|t) in eq 12, or equivalently how to predict Â fk,1(t + j|t) and Â fk,2(t + j|t), which is also the issue mentioned in eq 6. Suppose that ϕi:+ → + is a map from the extent of mold filled to cross-sectional area of the cavity. If i = 1, it means the filling extent is calculated from gate 1. A similar case holds for i = 2. Obviously, the following equation holds. ϕ1(s) = ϕ2(L − s),

∀ s ∈ [0, L]

Figure 5. Relation between Afk,i(t) and ϕi.

It is worth noting that despite the cost function defined in eq 12 being seemingly a quadratic form and solvable and the dynamics of the process being linear time invariant, it is a nonconvex cost function and even not differentiable because the term P̂ k(t + i|t) is related to ϕi, which is determined by the geometry of the mold cavity and is not necessarily continuous, not to mention differentiable. As for the prediction, Â fk,i(t + j|t) is highly dependent on the prediction of injection velocity of previous time instants. Thus, it is a highly nonlinear process, and so is the optimization problem, which is nonconvex as well. (This means that there may not exist a unique solution, and the problem cannot be solved in polynomial time.) To resolve the problem, the following two approaches are proposed to carry out the prediction of Â fk,i(t + j|t). ⎛ t yk , i (n) f Âk , i (t + j|t ) = ϕi⎜⎜ ∑ AbΔ f + Ak , i (n) ⎝ n=1

∀ s ∈ [0, L]

∑

A bΔ

n=t+1

yk − 1, i (n) ⎞ ⎟ f ⎟ ̂ Ak , i (n) ⎠ (16)

or

(13)

⎛ t yk , i (n) f Âk , i (t + j|t ) = ϕi⎜⎜ ∑ AbΔ f + Ak , i (n) ⎝ n=1

where L is the cavity length. If the cavity is symmetric for gates, then ϕ1(s) = ϕ2(s),

t+j−1

t+j−1

∑ n=t+1

A bΔ

yd, i (n) ⎞ ⎟ f ⎟ ̂ Ak , i (n) ⎠ (17)

(14)

The first approach uses the injection velocity of the same time instant in the last batch to predict the displacement in the future, while the second approach is much simpler, employing only the information involved in the desired velocity trajectory. No matter which approach is implemented, the value of Â fk,i(t + j|t) is

The function ϕi depends on only the mold design and is independent of the control algorithm. It is predetermined before design and implementation of any control law. It should be remarked that the ϕi-method allows complicated cavity 4799


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predetermined by recursive calculation before optimization operation. As for the term Σtn=1AbΔ(yk,i(n)/Afk,i(n)), there are some new measurement technologies that provide melt-front-position signal quantitively.30,31There is no doubt that this approach should be much more accurate. 3.2. Control Law Derivation. Without loss of generality, assume that prediction horizon is equal to control horizon, which is M = N. Then, rewrite eq 5 by stacking the velocity tracking error into a single vector.

Pk̂ ( tt ++ N1 |t ) = AbΔPk(t )1 + AbΔÂk ,1(t )[yd,1| tt ++ N1 f − ek̂ ,1( tt ++ N1 |t )] − AbΔÂk ,2 (t )[yd,2| tt ++ N1 − ek̂ ,2( tt ++ N1 |t )]

(25)

where 1 = [1

where

Pd| tt ++ N1

(19)

(20)

⎡Ci ⎤ ⎢ ⎥ ⎢ Ci ⎥ N Ci ≜ ⎢ ⎥ ⋱ ⎥ ⎢ ⎢⎣ Ci ⎥⎦

(21)

⎡ Bi ⎤ ⎢ ⎥ Bi ⎢ Ai Bi ⎥ N Bi ≜ ⎢ ⎥ ⋮ ⋱ ⎥ ⎢ ⋮ ⎢ A N − 1B A N − 2 B ... B ⎥ ⎣ i i i i i⎦ ⎡ ⎤ rk̂ , i(t |t ) ⎢ ⎥ ⎢ r ̂ (t + 1|t ) ⎥ k ,i ⎥ rk̂ , i( tt + N − 1|t ) ≜ ⎢ ⎢ ⎥ ⋮ ⎢ ⎥ ⎢⎣ rk̂ , i(t + N − 1|t )⎥⎦

ek − 1, i tt ++ N1

⎡ P (t + 1) ⎤ ⎢ d ⎥ ⎢ Pd(t + 2) ⎥ ≜⎢ ⎥ ⋮ ⎢ ⎥ ⎢ ⎥ ⎣ Pd(t + N )⎦

(26)

...

yd, i (t + N )]T

(27)

(28)

(29)

⎡ ⎤ 1 ⎢ f ⎥ ̂ ⎢ Ak , i (t + 1|t ) ⎥ ⎢ ⎥ 1 1 ⎢ ⎥ f f f ⎥ Âk , i (t ) ≜ ⎢⎢ Âk , i (t + 1|t ) Âk , i (t + 2|t ) ⎥ ⎢ ⎥ ⋮ ⋮ ⋱ ⎢ ⎥ 1 1 1 ⎢ ⎥ ... f f ⎢ ̂f ⎥ Âk , i (t + N |t ) ⎦ ⎣ Ak , i (t + 1|t ) Âk , i (t + 2|t ) (30)

This is followed by introducing the notations given below.

(22)

(23)

and ⎡e (t + 1) ⎤ ⎢ k − 1, i ⎥ ⎢e ⎥ k − 1, i(t + 2) ⎥, ≜⎢ ⎢ ⎥ ⋮ ⎢ ⎥ ⎢⎣ ek − 1, i(t + N )⎥⎦

yd, i (t + 2)

⎡ P ̂ (t + 1|t ) ⎤ ⎢ k ⎥ ⎢ P ̂ (t + 2|t ) ⎥ ⎥ Pk̂ ( tt ++ N1 |t ) ≜ ⎢ k ⎢ ⎥ ⋮ ⎢ ⎥ ⎢⎣ Pk̂ (t + N |t )⎥⎦

(18)

⎡ Ai ⎤ ⎢ ⎥ ⎢ Ai2 ⎥ N Ai ≜ ⎢ ⎥ ⎢ ⋮ ⎥ ⎢ N⎥ ⎣ Ai ⎦

... 1]1T× N

yd, i = [ yd, i (t + 1)

ek̂ , i( tt ++ N1 |t ) = ek − 1, i| tt ++ N1 − CiN A iN δxk , i(t ) − CiN BiN rk̂ , i( tt + N − 1|t )

⎡ e ̂ (t + 1|t ) ⎤ ⎢ k ,i ⎥ ⎢ e ̂ (t + 2|t ) ⎥ k , i ⎥ ek̂ , i( tt ++ N1 |t ) ≜ ⎢ ⎢ ⎥ ⋮ ⎢ ⎥ ⎢⎣ ek̂ , i(t + N |t )⎥⎦

1

⎡ e ̂ ( t + N |t ) ⎤ k ,1 t + 1 ⎥ ek̂ ( tt ++ N1 |t ) ≜ ⎢ ⎢ t+N ⎥ ⎣ ek̂ ,2( t + 1 |t )⎦

(31)

t+N ⎤ ⎡e k − 1,1| t + 1 ⎥ ek − 1| tt ++ 1N ≜ ⎢ ⎢ t+N ⎥ ⎣ ek − 1,2| t + 1 ⎦

(32)

⎡ δxk ,1(t )⎤ ⎥ δxk(t ) ≜ ⎢ ⎢⎣ δx (t )⎥⎦ k ,2

(33)

rk̂ ( tt + N − 1|t )

∀ i = 1, 2 (24)

A procedure similar to that for weld line position prediction, the position index prediction, can be depicted as follows according to eq 10. 4800

⎡ r ̂ ( t + N − 1|t )⎤ k ,1 t ⎥ ≜⎢ ⎢ t+N−1 ⎥ t | r ( ) ̂ ⎣ k ,2 t ⎦

(34)

⎡CN A N ⎤ 1 1 ⎥ Ã N ≜ ⎢ ⎢ C2N A 2N ⎥⎦ ⎣

(35)

⎡CN BN ⎤ 1 1 ⎥ ̃ BN ≜ ⎢ N N⎥ ⎢ C B ⎣ 2 2 ⎦

(36) DOI: 10.1021/ie5041397 Ind. Eng. Chem. Res. 2015, 54, 4795−4804

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Figure 6. Geometry of the mold cavity.

Therefore, the entire problem can be summarized as the

∂Jk (x) 2∂rk̂ ( tt + N − 1|t )

following optimization problem.

+ B̃ TN [Ã kf (t )]T R̂[Ωk(t ) − Ã kf (t )ek̂ ( tt ++ N1 |t )] = 0

min Jk (t ) = êkT( tt ++ N1 |t )Q̂ ek̂ ( tt ++ N1 |t ) + r ̂kT( tt + N − 1|t ) S r̂ k̂ ( tt + N − 1|t ) + [Ωk(t ) − Ã kf (t )ek̂ ( tt ++ 1N |t )]T R̂[Ωk(t ) − Ã kf (t )ek̂ ( tt ++ 1N |t )]

s.t.

=

ek − 1| tt ++ 1N

− Ã N δxk(t ) −

Ωk(t ) ≜ AbΔPk(t )1 +

Ã kf (t )yd| tt ++ N1

rk̂ ( tt + N − 1|t ) = {B̃ TN [(Ã kf (t ))T R̂ Ã kf (t ) + Q̂ ]B̃ N + S}̂ −1 {B̃ TN [(Ã kf (t ))T R̂ Ã kf (t ) + Q̂ ]ek − 1| tt ++ N1

B̃ N rk̂ ( tt + N − 1|t )

−

− B̃ TN [(Ã kf (t ))T R̂ Ã kf (t ) + Q̂ ]Ã N δxk(t )

Pd| tt ++ N1

−B̃ TN (Ã kf (t ))T R̂ Ωk(t )}

f f Ã kf (t ) ≜ [ AbΔÂ k ,1(t ) − AbΔÂ k ,2(t )]

yd tt ++ N1

(38)

Solve the above equation to obtain

rk̂ ( tt + N − 1| t )

ek̂ ( tt ++ 1N |t )

= − B̃ TN Q̂ ek̂ ( tt ++ N1 |t ) + S r̂ k̂ ( tt + N − 1|t )

(39)

The first and (N + 1)th components of r̂k(t t+ N−1|t) will be injected to actuators for implementation. It is noted that the framework can be easily extended to deal with control input limit by imposing such constraints into the optimization problem (eq 37). However, the closed form cannot be obtained in general. Before proceeding to the simulation for illustration, the uneven length issue should be addressed. The strategy for the proposed algorithm is to set up a buffer with length l¯ to store data, e.g., u1, u2, x1, x2 of the last batch. Once the length of the current batch exceeds that of the last batch, the controller can retrieve data from the buffer for prediction and calculation.

⎡ y |t+N ⎤ ⎢ d,1 t + 1 ⎥ ≜⎢ t+N ⎥ ⎢⎣ yd,2| t + 1 ⎥⎦

⎡Q ⎤ ⎢ ⎥ Q ⎢ ⎥ Q̂ = ⎢ ⋱ ⎥ ⎢ ⎥ Q⎦ ⎣ 2N × 2N ⎡R ⎤ ⎢ ⎥ R ⎥ R̂ = ⎢ ⋱ ⎥ ⎢ ⎢⎣ R ⎥⎦2N × 2N

4. NUMERICAL SIMULATION The dynamics of two injection units studied in simulation are as follows.8,27,32 Injection unit 1: ⎧ ⎡1.582 − 0.5916 ⎤ ⎡1 ⎤ ⎪ xk ,1(t + 1) = ⎢ ⎥xk ,1(t ) + ⎢⎣ ⎥⎦uk ,1(t ) ⎣ ⎦ 0 1 0 ⎪ ⎨ + dk(t ) ⎪ ⎪ yk ,1(t ) = [1.69 1.49]xk ,1(t ) ⎩

S ̂ = diag{S , S , ..., S}N × N (37)

where Ωk(t) represents the predicted error of weld line position.

(40)

Injection unit 2:

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Industrial & Engineering Chemistry Research ⎧ ⎡ 1.607 1 ⎤ ⎡ 1.2390 ⎤ ⎪ xk ,2(t + 1) = ⎢ ⎥x (t ) + ⎢ ⎥ ⎣− 0.9282 ⎦ ⎣− 0.6086 0 ⎦ k ,2 ⎪ ⎨ uk ,2(t ) + dk(t ) ⎪ ⎪ yk ,2 (t ) = [1 0]xk ,2(t ) ⎩

resulting velocity and control inputs associated with eq 17 are shown in Figures 9−12. It is noted that the fluctuation on velocity is normal because of the white-noise process disturbance. (41)

where dk(t) is process disturbance and is assumed to be subjected to normal distribution 5 (0.4,0.2). The mold geometry for simulation study is shown in Figure 6. The map of cross-sectional area versus displacement is demonstrated in eq 42. The crosssectional area of the barrel is 1000 mm2. The sampling time is 1 ms. The injection time is designed to be 1 s. Considering the smoothness of product surface, the melt front velocity should be even.33 Thus, the velocity profiles are designed as shown in Figure 7. The weights selected for cost function is Q = 1, R = 40, Figure 9. Velocity tracking of the first screw.

Figure 7. Velocity profile for both screws.

and S = 0.1. The prediction horizon and control horizon are both equal to 8. The first batch control is initiated with conventional MPC. ⎧ 400 ⎪ ⎪ 5(100 − s) ⎪ ϕ1(s) = ϕ2(s) = ⎨ 300 ⎪ ⎪ 5(s − 60) ⎪ ⎩ 400

Figure 10. Velocity tracking of the second screw.

s ∈ [0, 20) s ∈ [20, 40) s ∈ [40, 120) s ∈ [120, 140) s ∈ [140, 160]

(42)

4.1. Case 1: Without Model Mismatch. The result is shown in Figure 8. The red dashed line represents the prediction of cross-sectional area calculated from eq 16, while the green solid line is for eq 17. The performance of both are quite close, which implies that the control performance is robust to prediction error on cross-sectional area. Also, according to the result, the weld line positioning is controlled well and is improved gradually, compared to the conventional MPC. The

Figure 11. Control input of the first screw.

4.2. Case 2: With Model Mismatch. Considering the robustness of the proposed control law, the model for simulation

Figure 8. Performance comparison in terms of weld line position error.

Figure 12. Control input of the second screw. 4802


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is assumed as follows, which follows the standard way to check robustness of MPC.34,35 Injection unit 1: ⎧ ⎡1.682 − 0.5916 ⎤ M ⎪ xkM,1(t + 1) = ⎢ ⎥xk ,1(t ) ⎣ 1 0 ⎦ ⎪ ⎪ ⎡1 ⎤ ⎨ + ⎢ ⎥uk ,1(t ) ⎣0⎦ ⎪ ⎪ ⎪ ykM,1(t ) = [1.69 1.49 ]xkM,1(t ) ⎩

(43)

Injection unit 2: ⎧ ⎡ 1.507 1 ⎤ M ⎪ xkM,2(t + 1) = ⎢ ⎥x (t ) ⎣−0.6086 0 ⎦ k ,2 ⎪ ⎪ ⎡ 1.2390 ⎤ ⎨ +⎢ ⎥u (t ) ⎣−0.9282 ⎦ k ,2 ⎪ ⎪ ⎪ ykM,2 (t ) = [1 0]xkM,2(t ) ⎩

(44)

Figure 13 indicates that the conventional MPC is very sensitive to model mismatch in this case, as indicated by the large position

Figure 13. Performance comparison in terms of weld line position error.

error, while both proposed methods still achieve a precision with error less than 0.1 mm.

5. CONCLUSION A two-time-dimensional model predictive control has been proposed to solve the weld line positioning problem in the biinjection molding process. Two methods for cross-sectional area prediction are developed to reduce the complexity of the optimization problem. Simulations demonstrate good performance of the proposed methods and their robustness against model mismatch.

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AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS This project is supported in part by Hong Kong Research Grant Council, under Project 612512. REFERENCES

(1) Chen, Z.; Turng, L.-S. A review of current developments in process and quality control for injection molding. Adv. Polym. Technol. 2005, 24, 165−182. 4803


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