A Novel Predictive Function Control in Heating System of Injection Molding Machine

Studies in System Science (SSS) Volume 2, 2014 www.as‐se.org/sss

A Novel Predictive Function Control in Heating System of Injection Molding Machine Heng DU1, Jian‐ming DU*1, Nian‐sheng Xu2, Quan Long1, Heng‐zhi Cai2 1

College of Mechatronics and Control Engineering, Shenzhen University, Shenzhen Guangdong 518060, P. R. China

2

LK Leadwell Shenzhen Technology Co., Ltd., Shenzhen 518060, P. R. China

du123yiheng@qq.com; *djm33@szu.edu.cn; xns1959@sohu.com; 317970736@qq.com; caihzh@lklw.com.cn

Abstract This paper took charging barrel heating system of precision injection molding machine as study object, this work analyzed the major parameters affecting temperature control system by state space modeling. Global optimization performance of simulated annealing algorithm was used to improve the method of predictive function control in rolling optimization. Therefore, the defect that the objective function takes a direct derivative was avoided. Furthermore, predictive function control input variable was transformed into on‐off time variable control for thermocouple of charging barrel. The achieved control method, therefore, is closer to actual switch mode temperature control of the injection molding machine. Finally, this method was proved to be effective in decreasing the overshoot in control process, by actual test.

Keywords Predictive Functional Control; Simulated Annealing; Charging Barrel Temperature System Control; Temperature System Model

Introduction For precision injection molding, precise temperature control is one of the key factors affecting the quality of injection molded products. However, the object of charging barrel temperature control of injection molding machine is a multi‐variable, strong coupling, nonlinear, large inertia and time‐varying systems. What's more, charging barrel insulation resistance, melt material specific heat capacity, heating coil working power and the disturbance of external environmental temperature will affect the performance of the injection molding machine temperature control system. The injection molding machine temperature control method is more widely used PID in current. PID control method, although without considering the change of process model structure and parameters, but the traditional PID controller parameters are hard to tune timely under the rapid changes of the dynamic network circumstances [1].

Huailin Shu [2] proposed a PID Neural Network method to get perfect decoupling and self‐learning control performances.

Since the 1970s, the complex industrial process control challenges the modern control theory. The emergence of a large number of algorithms is to solve practical problems in industry, and the predictive control method is one of them. Bismark C. Torrico proposed a method based on predictive control and applied to a switched reluctance motor drive to achieve good relationship among robustness [3]. Predictive Functional Control (PFC) is characterized by simple, small calculation, fast and high precision tracking performance [4]. Zhihuan Zhang [5] proposed a new method to combine PFC with PID control, and obtained a better tracking features and robust performance in the hydraulic robot system control. Richalct also proposed a heuristic method established on the impulse response of non‐parametric model predictive [6] and successfully applied in the control of the distillation column. Calculation process of simulated annealing algorithm is simple, universal and robust, suitable for parallel processing, and can be used to solve complex nonlinear optimization problem [7‐9]. Therefore, this paper attempts to explore and combine the predictive control and simulated annealing algorithm to optimize control methods in order to achieve precision charging barrel temperature control of injection molding machine and to lay the foundation for the realization of precision injection molding.

Mathematical Model of the Object of Charging Barrel Temperature Control System In this paper, four heating zones of charging barrel of injection molding machine is taken into consideration, and they are nozzle heating zones, heating zones I, II and heating zone III respectively. Each zone equipped heating coils and can heat independently, as shown in Figure 1. The mathematical model of the heating regions is defined by Eq. (1), (2), (3) and (4).

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FIG. 1 THE CHARGING BARREL SCHEMATIC OF INJECTION MOLDING MACHINE

Nozzle heating zone: C0

dT0 (t )  Q0 (t )  [T0 (t )  Te ]r0  [T0 (t )  T1 ]r01 (1) d (t )

Heating zones I: C1

dT1 (t )  Q1 (t )  [T1 (t )  Te ]r1  [T1 (t )  T0 (t )]r01  [T1 (t )  T2 (t )]r12 (2) d (t )

C2

dT2 (t )  Q2 (t )  [T2 (t )  Te ]r2  [T2 (t )  T1 (t )]r12  [T2 (t )  T3 (t )]r23 (3) d (t )

Heating zones II:

Heating zones III: C3

dT3 (t )  Q3 (t )  [T3 (t )  Te ]r3  [T3 (t )  T2 (t )]r23 (4) d (t )

where Ti(t)(i=0,1,2,3), Ci(i=0,1,2,3), Qi(t) and ri(i=0,1,2,3) respectively denote the real‐time temperature of the nozzle, I, II and III of the heating zone, specific heat capacity of the melting material in the region, the heat generated by the heating coil and thermal conductivity coefficient between the charging barrel and the environment of the region. r01, r12 and r23 respectively denote thermal conductivity coefficient between the nozzle and the heating zone I, heating zone I and the heating zone II, the heating zone II and the heating zone III. Te denotes the environment temperature.

A state‐space matrix Eq. (5) can be obtained by simplifying and merging Eq. (1), Eq. (2), Eq. (3) and Eq. (4).  (r0  r01 )  C0  r01  dT0    dT1   C1  dT2    0  dT3     0  

r01 C0 (r1  r01  r12 ) C1 r12 C2 0

0 r12 C1 (r2  r12  r23 ) C2 r23 C0

  1  C   0  T0   0 0   T1    T2    r23  T3   0 C2   (r23  r3 )   0   C3  0

0

0

1 C1

0

0

1 C2

0

0

 r0   0  C   0   r1  0  Q0   C1    Q1   Q2   Te  r2  (5) 0  Q     3  C2   1   r3  C   C3   3

Formula (6) can be obtained by simplifying the mathematical model dT  AT  BQ  C (6)

where A, B, C respectively denote state‐space matrix, control matrix of its system and disturbance factor matrix to the charging barrel temperature. Seen from Eq. (5) and Eq. (6), heater input power and the temperature of charging barrel heating coil are positive correlation, and the main factor that influences the cylinder charging barrel temperature is each temperature zone to interact with its adjacent zone and the ambient temperature. The temperature of this interaction has a hysteresis characteristic; it’s the reason that makes the charging barrel temperature tend to converge. Thus each charging barrel zone temperature deliver is a mutual coupling and nonlinear interactions relationship.

Charging Barrel Temperature Predictive Control Algorithm Design Rules of PFC Algorithm The main advantage of PFC algorithm is that the output of the control model is seen as a response to some linear

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superposition of basic functions, and the predicted value ym(k+i) depends on the free response in past y0 and the control value in future yf, i.e. ym (k  i )  y0 (k  i )  y f (k  i ) (8)

Free response y0 depends on the amount of control in the last time. yn denotes the response value of the basic function, and we can obtain Eq. (9). y0 (k  j ) 

P 1

 yn (k  j  i) (9)

i  j 1

PFC regarded the control input u(k+i) as a known function that is several multiples of basic functions f(i), as Eq. (10) N

u (k  i )   n f n (i ), i  0,1, , P  1 (10) n 1

N

N

i 1

i 1

y f (k  i )   gi  u (k  i  1)   gi (  i 1 )  u (k ) (11)

where fn(i) denotes the value of the basic function in i‐th cycles, and N represents the number of basic functions. P is prediction horizon, and gi is a prediction model under the influence of the base sequence of output response function, and gi can be calculated through offline methods. Unknown input parameter u(k+i) is obtained by optimizing the minimum norm of forecast model output and the actual output difference. The value of the reference trajectory in the future i‐th time can be obtained by Eq. (12) yr (k  i )  a i yr (k )  (1  ai ) w(k  i ) (12) yr ( k )  y ( k )

where a  exp( Ts  ) , Ts denotes sampling time, and  denotes the time constant for the reference trajectory. w(k+i) denotes setting temperature in i‐th time, and y(k) is the output temperature value in current. Due to the presence of model errors, there need some correction for predicted output value in (k+i)‐th time, as show in Eq. (13) yc (k  i )  ym (k  i )  h  e(k ) (13) e ( k )  y ( k )  ym ( k )

where yc(k+i) denotes predicted output value after error compensation in (k+i)‐th time, and y(k) is actual output value. ym(k) denotes uncompensated output in k‐th time.

To achieve a dynamic optimal control for unknown complex model and to obtain the coefficients of linear combinations of basic functions forced response, we can establish an optimization objective function based on model predictive control, as show in Eq. (14) P

m

i 1

i 1

J   [ y r (k  i )  yc (k  i )]2     [u (k  i  1)]2 (14)

where yr denotes reference trajectory and yc denotes predicted values. P denotes Prediction horizon and  denotes weighting factor.

Based on Metropolis Acceptance Criteria Objective Function’s Optimization by Simulated Annealing Algorithm Let’s substitute Eq. (11) into Eq. (8), and then combined Eq. (12) substitute into the target Eq. (14). Minimize the objective function J to obtain the optimal control rate. For conventional PFC method to optimize the objective function, we often use the method that objective function derivative of the input variables equal zero, such as J   0 , but this method will most likely obtain the local minimum value or multiple solutions for objective function optimization; it can’t obtain the minimum when objective function is a non‐convex, non‐differentiable, shock changes function, so this method has some limitations. However, simulated annealing algorithm suitable for solving this problem and the algorithm is based on Metropolis acceptance criteria to avoid local optima. Adding simulated annealing algorithm in predictive control rolling optimization can make calculation more accurate.

To this end, let’s define the two states u(i) and u(j) at different input values, where i  N(i), N(i) is neighborhood of i.

The acceptance probability value of transition from the state i to state j is determined by Metropolis acceptance criteria as following, as shown in Eq. (15)

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1, f ( j )  f (i )   P （i  j）= exp(- J(u( j ))  J (u (i )) ), f ( j )  f (i ) (15)  kTsa

Where k is the Boltzmann constant and Tsa is simulated annealing temperature. Let’s generate a random number   U(0,1) to contrast with the value of P (i → j). When  ≦P(i→j), state change is accepted, otherwise, to go find another value which surrounding the neighborhood of i. When the number reaches the set parameter n(T(k)), the process proceeds to cool the annealing.

Simulated annealing process requires a high value of an initial annealing temperature Tsa0, and the annealing temperature gradually decreases, so the annealing temperature Tsa(k) is set to change the value of each outer loop. Its value satisfies the Eq. (16) Tsa (k )  Tsa 0 exp(ck 1 N ) (16)

where k denotes the number of outer loop, N is the number of parameters to be inversion and c is a constant. When Tsa(k) is lower than or equal to the final temperature Tf, the outer loop is terminated.

Conversion between Prediction Variables and Control Time For the general type injection molding machine, it often uses the heating coil temperature switch control method.

When heating is required, the solid state relay turns on; otherwise, the solid state relay is turned off. The control inputs only 0 or 1, and predictive control variable change its value with time, but they satisfy the same amount of heat generated in the same cycle TA. It can be converted by the following formula (17) t TA

U 02 [1u (t )]2  dt Ta (17)  R R t

where U0 is the voltage of the coil which is running at full power and R is the equivalent resistance of the coil. 1 is a scaling factor which equals to predictor variables divided by coil control voltage and Ta is the control time in control cycle TA. The smaller the value TA is, the more accurate the control action is; but the solid state relay switching between ON and OFF become more frequent.

Charging Barrel Temperature PFC Experiment In this study, this experiment adopts the codesys industry control software for the temperature acquisition and analysis, and the software data sampling period is 2s. With two minutes for a cycle to heat the charging barrel, the experiment was carried out by program. The nozzle, I zone, II zone and III zone charging barrel temperature output curves are as shown in Fig. 2.a, Fig. 2.b, Fig. 2.c and Fig. 2.d.

Conclusion Due to the small volume of the nozzle area, nozzle area temperature is more susceptible by ambient temperature, so it takes less time heated to a maximum temperature. As Fig. 2 shows, the solid black line represents the threshold control. In this segment curve indicate the coil is running at a full power heating, which could reduce the amount of computation of PID and PFC control. Table 1 which represents the performance parameters shows the PFC control in quick response and low overshoot control superior performance compared with PID control.

TABLE 1 EACH CHARGING BARREL REGION EXPERIMENT CONTROL PERFORMANCE PARAMETERS ABOUT PFC AND PID

NOZZLE ZONE

I ZONE

II ZONE

III ZONE

PFC

15.2

22.8

22.2

21.2

PID

18.95

25.92

25.26

23.06

PEAK TIME(MIN)

OVERSHOOT (%)

REGULATION TIME(MIN)

56

PFC

1.118

1.152

1

0.645

PID

1.706

1.576

1.75

1.871

PFC

13.8

21.2

20.6

18.4

PID

15.45

22.16

21.86

19.55

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a The curve of nozzle zone temperature output b The curve of I zone temperature output

c The curve of II zone temperature output d The curve of III zone temperature output FIG. 2 THE ACTUAL TEMPERATURE AND SETTING TEMPERATURE CURVE OF OUTPUT REFERENCES

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Jian‐ming DU (1963‐ ), Doctor, Professor of College of Mechatronics and Control Engineering, Shenzhen University, Shenzhen Guangdong 518060. P. R. China. His Research interests include system control and digital manufacturing.

Heng DU (1989‐ ), Postgraduate, College of Mechatronics and Control Engineering of Shenzhen University, Shenzhen Guangdong 518060. P. R. China. His Research interests include system control and process control.

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