injection molding control: from single cycle to batch control

Injection Molding Control: FromSingle Cycle to Batch Control

YI YANG, KE YAO, FURONG GAODepartment of Chemical Engineering, The Hong Kong University of Science & Technology,Clear Water Bay, Kowloon, Hong Kong

Received: 4 March 2009Accepted: 16 March 2009

ABSTRACT: Closed-loop control of key process variables in injection moldinghas become an essential part of the quality assurance system. Advanced controlalgorithms such as adaptive control and model predictive control have beenadopted to deal with the inherent process nonlinearity and time-varyingcharacteristics. These control algorithms are all focused on single-cycle controlperformance. Recently, a multicycle, two-dimensional model predictive learningcontrol has been developed for batch process control. This method has beenapplied to injection molding. In this paper, the basic principles of these controlalgorithms are introduced, with the control performance demonstrated byinjection velocity control. C© 2009 Wiley Periodicals, Inc. Adv Polym Techn 27:217–223, 2008; Published online in Wiley InterScience(www.interscience.wiley.com). DOI 10.1002/adv.20135

KEY WORDS: 2D system, Adaptive control, Batch process, Injection molding,Model predictive control

Introduction

A s a typical batch process, injection mold-ing consists of three main phases: filling,

packing-holding, and cooling. The molding pro-cess is a complicated multivariable process duringwhich the material variables, machine variables, pro-

Correspondence to: Furong Gao; e-mail: [email protected].

cess variables, and environmental variables inter-act with each other and determine the final productquality. With the continuous growth of the mold-ing industry and ever-expanding applications ofinjection-molded products, the demand for rapidproduction of complex parts with tighter tolerances,superior finish, and lower cost is increasing rapidly.These requirements can only be met with a so-phisticated process control system providing accu-rate control of key process variables that are inher-ently nonlinear and time varying. The complicated

Advances in Polymer Technology, Vol. 27, No. 4, 217–223 (2008)C© 2009 Wiley Periodicals, Inc.

INJECTION MOLDING CONTROL: FROM SINGLE CYCLE TO BATCH CONTROL

FIGURE 1. Batch process two-dimensional view.

dynamics of these variables have been demonstratedand studied extensively in some previous work.1−5

Advanced control strategies, such as adaptive con-trol and model predictive control (MPC), have there-fore been adopted to deal with the control problemby Gao and coworkers.3−5 These control strategies,however, are all focused on single batch performanceimprovement. In injection molding, which is a con-tinuous cycle batch process, the information of pastcycles can be exploited to improve the control per-formance of the current cycle.

A conventional control algorithm for a batch pro-cess is the iterative learning control (ILC), which ismotivated to mimic human learning. By using therepetitive nature of the process, other than improv-ing the control along the elapsed time index t duringa trial from step to step, ILC progressively and iter-atively improves the control accuracy along the trial

(or batch) index k from trial to trial. A conventionalILC scheme works as an open-loop feedforwardcompensator. It has been found that an ILC not us-ing within-cycle feedback tends to be both sensitiveto perturbation and slow in system convergence.6

For this reason, a real-time feedback control is of-ten combined with the conventional ILC, commonlyreferred as feedback–feedforward ILC,6 to improvecontrol performance not only along time but alsoalong cycle. The combined scheme tries to improvethe control accuracy along two time dimensions forcontrol input. This two-dimensional (2D) control re-sults in advantages over the conventional feedbackcontrol techniques where only one time-dimensionalinput action is made along the time axis. However,how to combine these two controls to ensure goodcontrol performance not only along the time direc-tion but also along the cycle direction is still an opentopic. In this paper, the ILC controlled batch processis modeled as a 2D system, as shown in Fig. 1, and a2D model predictive learning control algorithm hasbeen developed to optimize the control performancein both time direction and cycle direction, using in-formation not only in the current cycle but also inthe last cycle, as illustrated in Fig. 2.

These control strategies are briefly introduced inthe following sections, with the application demon-stration by injection velocity, a key variable in thefilling phase of injection molding.1,2,5 In the Con-trol Strategies section, the control background is de-scribed; in the Experimental section, the experimen-tal conditions used in this paper are given; in theResults and Discussion section, the control resultsand some discussion are presented; and finally, theconclusions are drawn.

FIGURE 2. Illustration of two-dimensional model predictive iterative learning control.

218 Advances in Polymer Technology DOI 10.1002/adv


FIGURE 3. Structure of an adaptive self-tuningcontroller.

Control Strategies

A brief outline of these control schemes is givenin this section for easy reference.

ADAPTIVE CONTROL

A self-tuning regulatory (STR) adaptive controlscheme has been successfully applied to injectionmolding control to cope with the nonlinear and time-varying characteristics of the process. The principlesand design procedure of the STR can be found inGao and coworkers.3,4,7 The STR shown in Fig. 3consists of two main loops, an ordinary feedbackcontrol loop, as shown in dashed line block, anda controller parameter adjusting loop, as shown inthe dotted line block. The latter consists of a para-metric model estimator and a controller design cal-culator, adjusting the parameters of the feedbackcontroller. The process model parameters and con-troller design are updated online during each sam-pling period, with a prespecified model structure.In the injection velocity adaptive controller design,a recursive least-squares method has been used forthe model identification and a pole-placement de-sign is adopted for feedback controller calculation.Adaptive feedforward control, an anti-windup esti-mation, and cycle-to-cycle learning have been devel-oped further to improve the control performance.2

GENERALIZED PREDICTIVE CONTROL

Generalized predictive control (GPC) design wasfirst proposed by Clarke et al.8 This method is robustin dealing with process uncertainties and model mis-match; therefore, it has been widely used in industry.

FIGURE 4. Illustration of model predictive control.

The overall structure of GPC is similar to that of STR,with only the GPC replacing the pole-placement de-sign. The basic idea of GPC is shown in Fig. 4. Onthe basis of an estimated process model, GPC de-termines an optimized sequence of future controlmoves to minimize a cost function defined over aprediction horizon, as follows:

J (N1, N2, Nu) = E

{N2∑

j=N1

δ( j)[y(t + j |t) − w(t + j)]2

+Nu∑j=1

λ( j) [�u(t + j − 1)]2

}(1)

where y(t + j |t) is the process output prediction overthe next j-steps at time t. N1, N2, and Nu are theminimum, the maximum, and the control horizons,respectively. δ and λ are the weighting factors. Inthis project, δ is set to be 1 and λ is set to be trace(G).With these assumptions, the controller output can bederived as follows:

u = −(GT G + λI)−1 GT (f − w) (2)

where u = [�u(t) �u(t + 1) · · · �u(t + Nu − 1)]T , Gis the process step response matrix, and w =[w(t + nd ) w(t + nd + 1) · · · w(t + nd + Nu − 1)]T isthe future set-point trajectory. The step response ma-trix G is allowed to be updated online by the real-time estimation of the process dynamics so that thenonlinearity and time-varying characteristics of theprocess can be traced.

Advances in Polymer Technology DOI 10.1002/adv 219


TWO-DIMENSIONAL MODEL PREDICTIVEITERATIVE LEARNING CONTROL(2D-MPILC)

Consider the following discrete input–outputbatch process model:

�BP : A(q−1)yk(t) = B(q−1)uk(t) + wk(t) (3)

where k is the cycle number, t is the sampling time,y is process output, u is the process input, thatis, the controller output, and A(q−1) = 1 + a1q−1 +a2q−2 + · · · + anq−n and B(q−1) = b1q−1 + b2q−2 +· · · + bmq−m are the input and output polynomials,respectively. Using the following ILC law to controlprocess (3),

uk(t) = uk−1(t) + rk(t), u0(t) = 0, t = 1, 2, . . . , T(4)

where rk(t) is the learning function to be determinedand u0(t) is the initial values of controller output. Thesystem described by Eqs. (3) and (4) can be viewedas an equivalent 2D system,

�2D : A(q−1)yk(t) = B(q−1)rk(t)

+ A(q−1)yk−1(t) + �wk(t) (5)

where �wk(t) = wk(t) − wk−1(t) is the nonrepetitivenoise. The model prediction in the time and cycledirections can be obtained as follows:

yk|k(∣∣t+1

t+n1

∣∣t) = Grk

(∣∣tt+n2−1

)+ yk−1

(∣∣t+1t+n1

)+ Fk(t)

(6)

yk+l|k(∣∣t+1

t+n1

∣∣t) = Grk+l

(∣∣tt+n1−1

)+yk+l−1|k

(∣∣t+1t+n1

∣∣t) , l > 0 (7)

Fk(t) can be calculated using the previous inputsand outputs.

According to this output prediction, the cost func-tion can be written as follows:

J (t, k, n1, n2, n3) =n3∑

l=1

λ(l)

(n1∑

i=1

α(i)(ek+l−1|k(t + i |t))2

+n2∑

j=1

β( j)(rk+l−1(t + j − 1))2

)(8)

where ek+l|k(t + i |t) = yr (t + i) − yk+l|k(t + i |t) is theerror of model prediction. The controller design is to

find out optimal learning action, rk(t), to minimizecost function (8). Because of space limitations, thedetailed derivation is not included in this paper. Thecalculated rk(t) can be shown as follows:

r∗k

(∣∣tt+n2−1

)= (

λ(1)R + GT Q1G)−1

GT Q1

×(

ek−1

(∣∣t+1t+n1

)− Fk(t)

)(9)

Matrix Q1 can be inversely calculated iteratively as,

Pn3 = 0, Ql = λ(l)Q + Pl (10.1)

Pl = Ql+1 + Ql+1G(λ(l + 1)R + GT Ql+1G)−1GTQl+1

(10.2)

where l = n3, n3 − 1, . . . , 1.Denoting the first row of matrix (λ(1)R +

GT Q1G)−1GT Q1 in Eq. (9) as K , the following con-trol law can be derived:

�2D−MPILC : uk(t) = uk−1(t) + K(

ek−1

(∣∣t+1t+n1

)− Fk(t)

)(11)

Equation (11) is the control law of 2D-MPILC; itcan be expressed in another way for easy under-standing,

uk(t) = uilc,k(t) + umpc,k(t)(12.1)

�ILC : uilc,k(t) = uilc,k−1(t) + K1ek−1

(∣∣t+1t+n1

)(12.2)

�MPC : umpc,k(t) = K2yk

(∣∣t−n+1t

)+ K3uk

(∣∣t−m+1t−1

)(12.3)

It is clearly shown in Eq. (12) that the control actionis divided into two parts, the ILC part and the MPCpart. These two controls have been combined in asystematic way.

Experimental

The machine used in this work is a ChenHsong reciprocating screw injection molding ma-chine (model number JM88MKIII). The maximummachine clamping force is 88 ton and the maximumshot weight is 128 g. A Temposonics transducer(RH-N-0200M) is used to measure both the screw



FIGURE 5. Self-tuning regulatory controller response to a step-change profile with different barrel temperatures.

position and the injection velocity. For the ex-periments described in this paper, high-densitypolyethylene and a mold with a cup-shaped cavitywere used.

Results and Discussion

These control algorithms are implemented in theinjection molding process. The injection velocity, atypical molding variable, is chosen to demonstratethe controller performance. The detailed analysis ofinjection velocity dynamics can be found in Gao andcoworkers.1,2

ADAPTIVE STR CONTROL RESULTS

The STR adaptive control is first implementedto control the injection velocity. Figure 5 illustratesthe response of the self-tuning controller with apole-placement control design to a step-down set-point profile for different barrel temperatures. Itcan be seen that the controller worked well. No-tice that the good control performance is achievedwith the enhancements of an adaptive feedforwardcontrol to speed up the step response time, an anti-windup estimation to eliminate the windup problemin model identification, and cycle-to-cycle learningto improve the control in the initial injection stage.

The STR controller has been also tested with differ-ent molds of various shapes and geometries and dif-ferent materials with various properties. Consistentand good performances have been achieved in allcases.2 The adaptive controller is therefore suitablefor injection velocity control. However, it is usualto have mismatch between the order of the modeland of the real process. This model order mismatchcan deteriorate the adaptive control performance toa certain degree. A wrong model order could leadto poor control performance when using adaptivecontrol. More detailed information can be found inYang and Gao.5

GPC CONTROL RESULTS

A GPC controller is also designed and tested ex-perimentally for the injection velocity. The adaptiveGPC responses to different barrel temperatures areplotted in Fig. 6. It shows that GPC provides perfor-mance similar to that of the STR. Different conditionshave also been tested and the GPC showed good per-formance over a wide range of operating conditions.It must be pointed out that the GPC has an inherentgood performance in set-point profile tracking, dueto the prediction and receding optimization. It hasalso been proved that the GPC is more robust againstmodel mismatch, even if the model structure is notidentified well, the GPC still can provide good con-trol performance.5



FIGURE 6. Generalized predictive control response to a step-change profile with different barrel temperatures.

2D-MPILC RESULTS

A 2D-MPILC controller is also designed for theinjection velocity. Velocity control simulation resultsare shown in this paper. The velocity model is iden-tified using open loop test results and formulated asfollows:

yk(t) = 2.651q−1 + 5.298q−2 + 0.5805q−3

1 − 1.454q−1 + 0.5285q−2 − 0.04736q−3 uk(t)

(13)

A lower order model is adopted for the con-troller design to test the controller robustness againstmodel mismatch,

yk(t) = q−1 13.81q−1

1 − 0.9524q−1 uk(t) (14)

Figure 7 shows the step responses of models (13)and (14); it is obvious that there exists significantmodel mismatch. The control results of 2D-MPILCare shown in Fig. 8. As within-cycle feedback controlis included, the control performance in the steadystage of the first cycle is reasonably good, as shownby the dotted line in Fig. 8. However, the initial con-trol performance as shown in the zoomed-in plot andthe transient behavior around the step-change pointare still not good. This is actually the problem of tra-ditional single-cycle feedback control. With ILC, thecontrol performance, especially the performance in

the initial filling stage, and step response are signif-icantly improved in the second and third cycles, asshown by the dashed line and the dash-dot line inFig. 8, respectively. The controlled injection velocityof the third cycle has already been close to the set-point profile. The solid line in Fig. 8 shows the 30thcycle’s control performance where the velocity over-laps the set-point profile, indicating superior controlperformance.

FIGURE 7. Comparison of step responses of processand model.



FIGURE 8. Injection velocity control using two-dimensional model predictive iterative learning control.

Conclusions

Because of the complicated characteristics of in-jection molding variables, it is necessary to applyadvanced control algorithms in injection molding.The traditional feedback control such as self-tuningadaptive control and adaptive GPC control areall focused on single-cycle control performanceimprovement. A 2D-MPILC method proposed inthis paper can improve the control performancein both the time direction and the cycle direc-tion and hence can be applied for batch pro-cess control. Applications in injection velocity con-trol have proved the effectiveness of this controlmethod.

References

1. Tsoi, H. P.; Gao, F. Polym Eng Sci 1999, 39, 3.2. Yang, Y.; Gao, F. Int Polym Process 1999, XIV, 196.3. Gao, F.; Patterson, W. I.; Kamal, M. R. Polym Eng Sci 1996, 36,

1272.4. Gao, F.; Patterson, W. I.; Kamal, M. R. Polym Eng Sci 1996, 36,

2467.5. Yang, Y.; Gao, F. Control Eng Pract 2000, 8, 1285.6. Bien, Z. Z.; Xu, J.-X. Iterative Learning Control: Analysis,

Design, Integration and Applications; Kluwer Academic:Boston, MA, 1998.

7. Astrom, K. J.; Wittenmark, B. Adaptive Control, 2nd ed.;Addison-Wesley: Boston, MA, 1995.

8. Clarke, D. W.; Mohtadi, C.; Tuffs, P. S. Automatica 1987, 23,137.


injection molding control: from single cycle to batch control

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