Self-optimizing MPC of melt temperature in injection moulding

ISA TRANSACTIONS® ISA Transactions 41 共2002兲 81–94

Self-optimizing MPC of melt temperature in injection moulding R. Dubay* Department of Mechanical Engineering, The University of New Brunswick, PO Box 4400, Fredericton, New Brunswick, Canada E3B 5A3

共Received 7 June 2000; accepted 16 February 2001兲

Abstract The parameters in plastic injection moulding are highly nonlinear and interacting. Good control of plastic melt temperature for injection moulding is very important in reducing operator setup time, assuring consistent product quality, and preventing thermal degradation of the melt. Step response testing was performed on the barrel heating zones on an industrial injection moulding machine 共IMM兲. The open loop responses indicated a high degree of process coupling between the heating zones. From these experimental step responses, a multiple-input–multiple-output model predictive control strategy was developed and practically implemented. The requirement of negligible overshoot is important to the plastics industry for preventing material overheating and wastage, and reducing machine operator setup time. A generic learning and self-optimizing MPC methodology was developed and implemented on the IMM to control melt temperature for any polymer to be moulded on any machine having different electrical heater capacities. The control performance was tested for varying setpoint trajectories typical of normal machine operations. The results showed that the predictive controller provided good control of melt temperature for all zones with negligible oscillations, and, therefore, eliminated material degradation and extended machine setup time. © 2002 ISA—The Instrumentation, Systems, and Automation Society. Keywords: Predictive control; Injection moulding; Self-optimizing; Material properties

1. Introduction In the overall plastic injection moulding process, the control of the machine process parameters is a challenge to researchers and end-users since several interactions exist between these machine parameters. The control of melt temperature is considered to be most critical factor in injection moulding since this parameter directly affects the melt viscosity of the polymer. Variations in melt temperature and hence melt viscosity have been shown to directly affect the translational injection velocity, the rotational reciprocating screw speed, the mould cooling duration, and the cavity *Tel.: ⫹1-506-458-7770; fax: ⫹1-506-453-5025; E-mail address: [email protected]

pressure-time profile 关1兴. On an industrial injection moulding machine 共IMM兲 screw barrel, the number of melt temperature zones on is generally greater that two excluding the nozzle. Basic on-off and proportional control have been used widely for barrel temperature control. Significant work has been done in modeling, simulation, and control of injection moulding parameters using conventional proportional, integral and derivative 共PID兲 based methods 关2–9兴. These methods provided limited control performance ranging from reasonable to unsuitable since they all lacked the robustness and adaptability necessary for controlling parameters that are nonlinear and coupled. Good steady-state temperature responses were achieved only when extensive gain scheduling was applied and controller parameters were redefined

0019-0578/2002/$ - see front matter © 2002 ISA—The Instrumentation, Systems, and Automation Society.

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Nomenclature

d N n1 n2 nu pk u u⫺ w(k) y yˆ y sp

process delay time process prediction horizon lower limit of prediction horizon upper limit of prediction horizon controller horizon step response coefficients controller output previous controller output weights on controller change in objective function J process output predicted process output setpoint for process output

Greek ⌽

adjustment variable for predicted profile ␤ parameter for weights on predicted errors ␣ parameter for modified setpoint profile ␦(k) weights on predicted errors in objective function J ⌬u controller output increment ⌬u ( t⫹k ) controller change at instant ( t⫹k ) evaluated at instant t

for several operating bands. However, poor transient responses resulted when using these conventional controllers. Model predictive control 共MPC兲 has been widely utilized for controlling chemical and petrochemical processes quite successfully. Its attractiveness as a control method is based on the ability to utilize an explicit plant model or process dynamic matrix obtained from opened loop step or impulse response testing, for predicting the effect of control actions on the controlled variables. The controller moves are determined by minimizing the difference between the desired and predicted response trajectories at every sampling instant. Constrained or unconstrained optimization techniques have been used in predictive control methodologies 关10,11兴. This general concept has been applied to several plants that are multivariable and

have a high degree of nonlinearity: a fluid catalytic cracker 关12,13兴, a highly nonlinear batch reactor 关14兴, and a hydrocracker 关15兴. More recently, Haung et al. 关16兴 used an adaptive generalized predictive control 共GPC兲 strategy for control simulations of plastic melt temperature in injection moulding. The GPC method was developed by Clarke and Scattolini 关17兴 and worked effectively for nonminimum phase processes with deadtime or time-delay. Haung et al. 关16兴 utilized a SISO 共single-input–single-output兲 second-order plus time delay model 关2兴 with variable process gain and time constants for the control simulations. The result of these simulations indicate temperature overshoot magnitudes ranging from 7 to 12 °C for a positive setpoint perturbation of 10 °C and temperature undershoot magnitudes ranging from 3 to 5 °C for a negative setpoint change of 5 °C. In some instances, the GPC temperature responses initially increased dramatically for a decreasing or negative setpoint change. Haung et al. 关16兴 did not investigate the control of the other melt temperature zones and the effect of their interactions, and therefore, the GPC controller was a SISO type. However, Haung et al. 关16兴 demonstrated that the single temperature zone is at least a second-order system with time delay, and that the process gain, time delay, and time constant are all input dependent. This essentially means that the temperature variable is highly nonlinear and that conventional control methods will only provide reasonable control performance within a narrow operating band, specific to the plastic material being processed. The development of PID gain scheduling using fuzzy logic by Blanchett et al. 关18兴 was used for controlling the temperature of an aluminum cylindrical block. The temperature responses when using the fuzzy PID controller performed well in comparison to the ones obtained using a MPC strategy. The fuzzy controller design was based on zone of heating and therefore its performance with multiple heating zones was not investigated. A typical IMM setup is to have a temperature gradient of 150– 200 °C 关19兴 from the first zone where the polymer enters the barrel to the nozzle where the molten plastic enters the mould. Due to this temperature gradient and the need to maximize the efficiency of the melting process for the plastic, it is imperative that a controller for melt temperature be designed to account for process in-

R. Dubay / ISA Transactions 41 (2002) 81–94

83

Fig. 1. Interaction of process variables in injection moulding. Fig. 2. General MPC method.

teractions resulting from energy flow between the zones as shown in Fig. 1, and provide a generic learning and self-optimizing approach for different polymers and IMM’s. Also, the controller should be able to guide the process response to achieve minimum rise time and yet have negligible or zero oscillations. These control performance requirements provide the research incentives for the investigations in this paper. To reduce the computational effort especially for realtime control of the parameters in injection moulding, which can have time constants in the order of 5 ms or less 共control of injection speed, back-pressure, and screw recovery speed兲, an unconstrained optimization approach will be used. A MIMO predictive controller will be developed and practically implemented for controlling the three temperature zones on an industrial IMM.

tive function subjected to constraints in the magnitude of the control moves and its changes. The result of the optimization procedure is a vector ⌬u that contains the current and future control changes in the manipulated variables that will be used to input to the plant. The MPC method uses two projections into the future for evaluating the process outputs and control inputs to the plant as shown in Fig. 3. The prediction of the process is evaluated over a horizon N. This length of this horizon is based on an open loop test using a step, multistep, or impulse inputs to the process that is to be controlled. The control changes in the manipulated variable that

2. Theory The general method of model-based predictive control is illustrated in Fig. 2. The approach uses a process model, and the past and future control moves to determine a process predicted trajectory. Past process outputs are used to evaluate an adjustment parameter that redefines the predicted output trajectory. At any instant t, the past process outputs are the discrete values of the process or controlled variable, previous to t. The future errors are calculated as the difference between the discrete setpoint trajectory that is previously evaluated, and the discrete process predicted values after time t. These errors are minimized by the optimizer block that is defined by a cost or objec-

Fig. 3. The control and prediction horizons.

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constitute the ⌬u are evaluated over a control horizon n u . The number of changes evaluated in ⌬u is generally less than 3 and, therefore, represents the vector length. The control horizon n u is the number of control moves changes or deviations in the manipulated variable that are calculated in order to affect the predicted outputs over the prediction horizon N. As demonstrated in 关20兴, a larger value of the control horizon n u results in the larger predictions of the process and consequently, smaller future errors to be minimized. This translates into larger control changes to be implemented to the controlled process at the current time instant t. In this paper, the MPC methodology for controlling the barrel and nozzle melt temperatures uses a two band 共normal IMM operation兲 MIMO approach for evaluating the magnitude of the control actions and for predicting the process trajectories. For a multivariable linear system having m controller inputs with its j process outputs that can respond to each of the m inputs, the output response for any jth process at any discrete time t within an operating band or range can be expressed as b

b

m

y r, j 共 t 兲 ⫽y r, j 共 0 兲 兺

N

兺

q⫽1 k⫽1

mized or avoided. Since the dynamic matrix pb comprising the discrete response coefficients can be formulated for several bands of operation, the nonlinearity behavior of the processes is readily incorporated into the controller design. The derivation of mathematical models in 关21兴 incorporated the process interaction effects in injection moulding. While these models are very useful for understanding the mechanisms involved, the use of these exhaustive models in the design of the controller is difficult since there are numerous variables that are unknown or not quantifiable. Generally, these unknowns are factors that involve material properties and process nonlinearities and therefore would present a formidable challenge on using conventional control schemes for controlling melt temperature, injection speed, and the like. The model-based predictive control scheme in this study uses a process dynamic matrix pb that includes 共1兲 the effects of the other processes and hence its inputs, 共2兲 the operating region for each of the j processes, and 共3兲 the rhelogical characteristics of the polymer. The superscript b on the pb matrix represents a set of elements specific to an operating region. 2.1. Objective function

b,q

p r,k, j ⌬u 共 t⫺k 兲 . q

共1兲 Eq. 共1兲 characterizes an open-loop MIMO step response model for any process j, based on the effect of q input changes within an operating band b. The magnitude of the discrete step response b,q

coefficient p r,k, j for input q takes into account the material properties r, a desired operating range b, and the power rating of the heaters on any IMM at any discrete instant k. The variable b

y r, j ( 0 ) is the initial steady state or magnitude for process j. The formulation of a MIMO model therefore comprises these discrete step response coefficients for each j process affected by each of the m inputs into a dynamic matrix pb , corresponding to an operating region b and polymer type r. This general approach is essential in the injection moulding of high precision plastic components since all the processes to be controlled are highly nonlinear and interactive, and timely machine setup and material wastage are to be mini-

The general aim of model-based predictive control methods is to minimize a vector of predicted errors e obtained by the difference between a reference trajectory yr,j and a predicted process trajectory yˆ r, j over a considered prediction horizon N. The control effort ⌬u necessary for doing so is evaluated through the constrained or unconstrained minimization of an objective function. This objective function can be expressed as

再兺 n2

J 共 n 1 ,n 2 ,n u 兲 ⫽Min b

k⫽n 1

b

␦ 共 k 兲关 yˆ r, j 共 t⫹k 兩 t 兲 nu

⫺y r, j 共 t⫹k 兲兴 2 ⫹ 兺 w 共 k 兲 k⫽1

冎

⫻ 关 ⌬u 共 t⫹k⫺1 兲兴 2 .

共2兲

The notation ( t⫹k 兩 t ) indicates the value of the variable predicted at instant ( t⫹k ) calculated at time instant t.

R. Dubay / ISA Transactions 41 (2002) 81–94

The parameters n 2 and n 1 are the maximum and minimum limits of the prediction horizon N, and the control horizon n u . The variables n 2 and n 1 identify the limits of the instants for which it is desirable for the output to follow the reference trajectory. Intuitively, if a large value of n 1 is selected, the inference is that errors in the initial or early instants are unimportant. Note that in processes with dead time d, the output u( t) will not affect the process until ( t⫹d ) . Therefore, n 1 and n 2 can be initialized as

n 1 ⫽d⫹1,

共3兲

n 2 ⫽d⫹N.

共4兲

The notation yˆ ( t⫹k ) is a k -step prediction of the process output at time t. The coefficients ␦ ( k ) and ␻ ( k ) are weighting sequences that can be imposed on the vectors of predicted errors and control moves. These coefficients can be constant or exponential sequences. If an exponential weighting sequence is used, then an expression for either of these weight factors can be expressed as

␦共 k 兲⫽␤

n 2 ⫺k

0⬍ ␤ ⬍1

b,l

¯

0

p r,1,2

p r,1,1

0

]

]

¯

]

l

b,l

b,l

b,l

p r,1,N

p r,1,N⫺1

b,l

0

p r, j,2

p r, j,l

0

]

]

¯

]

b,l

b,l

p r, j,N

b,l

p r, j,N⫺1

effect on process 1 due to input m

effect on process m

step input m

due to input 1

to process m

effect on process j

effect on process j

due to input 1

due to input m

0

¯

0

p r,1,2

p r,1,1

0

]

]

¯

p r,1,1 b,m

p r,1,N

b,l

p r, j,N⫺n u ⫹1

]

] b,m

b,m

p r,1,N⫺1

¯

p r,1,N⫺n u ⫹1

0

¯

0

p r, j,2

p r, j,1

0

]

]

¯

p r, j,1 b,m

]

¯

冤

step input 1 to process 1

b,m

¯

b,l

pb ⫽

b,m

p r,1,N⫺n u ⫹1

0

p r, j,1

As stated earlier, the formulation of the MIMO model is embedded in the dynamic matrix pb , which contains the normalized step response coefficients of the j process outputs that were obtained by introducing m step inputs acting on the j processes to be controlled. For any polymer and IMM, the MIMO responses that form the dynamic matrix pb can be represented in Eq. 共6a兲 with the coefficients shown in Eq. 共6b兲,

b,m

]

¯

2.2. The MIMO model

共5兲

b,l

b,l

In the case of ␦ ( k ) , if ␤ lies between 0 and 1 then the errors farthest from instant t are weighted more than those near instant t. This provides smoother control with less control effort. Conversely, if ␤ is greater than 1, the initial errors are weighted more providing tighter control and larger magnitudes of the control effort.

b,m

0

p r,1,1

pb ⫽

␤ ⬎1.

or

85

b,m

b,m

p r, j,N

b,m

p r, j,N⫺1

]

] b,m

¯

p r, j,N⫺n u ⫹1

m

.

冥

,

共6a兲

共6b兲

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R. Dubay / ISA Transactions 41 (2002) 81–94

In the MIMO dynamic matrix pb , the first subscript r represents the polymer type, the second indicates which of the j processes was subjected to the step input, and the third represents the time instant or location on the prediction horizon N. The first superscript on the elements represents the operating region or band b, and the second represents which of the m inputs was used to generate the corresponding step response coefficients. In this matrix, the step response coefficients are directly affected and hence the control move changes ⌬u by any deviations in the polymer thermophysical properties, the region of operation, and the power capacity of the electrical heaters. It is now clear that the thermophysical properties of the polymer, the IMM heating capacity for each zone of heating, and the temperature operating range can be modeled in the dynamic matrix pb . The dimension of pb is 兵 m ( N⫺n u ⫹1 ) by j M 其 . 2.3. Process prediction and adjustment The prediction horizon N is the number discrete sampling instants that all the melt temperature trajectories are predicted into the future. In this investigation, the control horizon n u is the same for the m manipulated variables and the prediction horizon is the same for the j process outputs. The control horizon has a direct influence on the magnitude of the discrete process predictions yˆ ( t ⫹k 兩 t ) and hence the size of the control moves changes ⌬u. A larger value of n u results in larger magnitudes of the predicted horizon. This results in smaller discrete values of the predicted errors and hence larger control changes to be made at current time to obtain these smaller future errors. Using these two horizons and the MIMO dynamic matrix pb , a general prediction of each process output at any time index ( t⫹k ) can be expressed as

yˆ r, j 共 t⫹k 兩 t 兲 b m

b

⫽yˆ r, j 共 t 兲 ⫹ 兺

k

兺 q⫽1 i⫽1

m

⫹兺

N

b,q

p r, j,i ⌬uq 共 t⫹k⫺i 兩 t 兲

b,q

b,q

兺 共 p r, j,i ⫺ p r, j,i⫺k 兲 q⫽1 k⫽i⫹1 b

⫻⌬uq 共 t⫹k⫺i 兲 ⫹ ␾ r, j 共 t 兲 .

共7兲

Fig. 4. Adjustment of the predicted profile.

b

The first term yˆ r, j ( t ) is the current prediction for process j using plastic type r within operating region b, the second contains the effect of the current and future control actions on the predicted process trajectory, and the third contains the effect of past control moves. In order to compensate for external disturbances, modeling mismatch and b

nonlinearities, an adjustment parameter ␾ r, j ( t ) is added or superimposed on each process prediction profile over the prediction horizon N. This adjustment is calculated from the difference between the instantaneous process measurement and the process prediction for time t made at time index ( t ⫺1 ) . Fig. 4 illustrates this calculation as time recedes from t to ( t⫺1 ) , b

b

b

⌽ r, j 共 t 兲 ⫽y r, j 共 t 兲 ⫺yˆ r, j 共 t 兩 t⫺1 兲 .

共8兲

For convenience, the superscript b will be neglected from now on. 2.4. The modified and the reference trajectories Research on the use of constrained optimization methods in predictive control for determining the optimal control move vector has been limited to simulations and processes with relatively large time constants 共greater than 1 s兲 such as in petrochemical processes. In plastic injection moulding, good control of process parameters such as injection speed, cavity pressure, and plastication backpressure rely on sampling intervals of less than 5 ms 关21兴. Since the application of constrained optimization methods in predictive control is computationally intensive, controlling these injection

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87

Fig. 6. Reevaluation of the modified setpoint profile. Fig. 5. The modified setpoint profile.

moulding parameters by these methods is limited. This paper addresses this problem by using a generic formulation for a modified setpoint profile 共MSP兲 v j ( t ) using a reference trajectory or setpoint s j ( t ) and the measured process y j ( t ) in order to reduce the dependence on constrained optimization, and yet achieve good control performance and simplified tuning. The model for defining v j ( t ) is typical of a second-order model with critical damping 关22兴 initiating from the value of the process y j to be controlled towards a known desired setpoint s j ( t ) . The general form of v j ( t ) over the prediction horizon N is can be expressed as v j 共 t⫹k/t 兲 ⫽ ␣ j v j 共 t⫹k⫺1 兲

⫹ 共 1⫺ ␣ j 兲 s j 共 t⫹k 兲 v j 共 t 兲 ⫽y j 共 t 兲 , ᭙k⫽1,...,N.

point s j more steeply, thereby increasing the aggressiveness of the controller. A key ingredient is that the MSP v j which spans the prediction horizon N is recalculated every sampling interval ( ᭙k⫽1,...,N ) initiating from the current measured value of each process. Fig. 6 shows the revaluation of v j ( t ) from the measured value of the process y j ( t ) . Also, the reevaluation of the MSP approaches the measured process output and finally converges to the setpoint. 2.5. Implementation of control moves For each zone temperature, the vector of predicted errors e is evaluated as the difference between the desired and the adjusted predicted temperature profiles. The individual predicted errors e i j and the error vector e are evaluated as

共9兲

The variable s j represents the desired operating magnitude for each of the j processes. Ohshima et al. 关23兴 showed that the parameter ␣ j provides a more direct and intuitive tuning parameter to the magnitude of ⌬u, than factors such as weighting sequences in the objective function and horizon lengths, as compared to the computationally intensive constrained optimization techniques. Also, the robustness of the predictive controller is improved and closely linked to the tuning parameter ␣ 关23兴. Fig. 5 shows the modified setpoint profile in relation to the parameter ␣. It is clear from Eq. 共9兲 that as ␣ decreases, the MSP approaches the set-

e ji ⫽ v ji ⫺yˆ ji ,

i⫽1,...,N,

共10兲

e⫽ 关 e 11 e 12¯e 1N ¯e j1 e j2 ¯e jN 兴 T . 共11兲 The first subscript on the error vector is the process 1,..., j, and the second is the position on the prediction horizon. The control or manipulated variable changes for each process are obtained by performing the following computations:

⌬u⫽ 共 pT p⫹Iw 兲 ⫺1 pT 共 v ⫺yˆ 兲

共12兲

⌬u共 t⫹k⫺1 兲 ⫽0,

k⬎n u ,

共13兲

⌬u共 t⫹k⫺1 兲 ⫽0,

k⭐n u .

共14兲

with

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R. Dubay / ISA Transactions 41 (2002) 81–94

The ⌬u vector is for all m manipulated variables and is evaluated as ⌬u⫽ 关 ⌬u1,1⌬u1,2¯⌬u1,n u ¯⌬um,1⌬um,2¯⌬um,n u 兴 T . 共15兲

The first control move changes are implemented to each process and the control calculations are repeated at the next control instant. Therefore, the set 兵 ⌬u11⌬u21⌬u31其 are the changes in the controller output or manipulated variable to the corresponding to processes j⫽1,2,3. The remaining elements of the ⌬u vector are used for evaluating the prediction trajectories of each process. The purpose of inputting the first control move changes and not the remaining ( n u ⫺1 ) is to enable the MPC algorithm to act on disturbances that may have entered the system between process sampling instants. Also, better correction of the process prediction is achieved since the adjustment factor ␾ 关see Eq. 共8兲兴 can be applied every sampling instant thereby reducing mismatch between the actual process and the model obtained by opened-loop testing. The manipulated variables to be applied to the each barrel temperature zone are evaluated as

u⫽u⫺ ⫹⌬u.

共16兲

3. Manipulated variables and controller parameters In this section, the manipulated variables and the controller parameters are specified for both MIMO and SISO predictive control. 3.1. Manipulated varaibles For SISO and MIMO predictive control of melt temperature on each zone, the energy provided by each electrical heater is the manipulated variable for controlling temperature. There are three zones of heating corresponding to two heater bands rated at 1100 W each, and the nozzle at 300 W. The electrical energy was manipulated by utilizing the time proportioning principle using three solid state relays 共SSR’s兲 having zero crossover switching, which greatly reduces radio frequency interference and switching voltage transients. The line voltage applied to the heaters was 220 V ac at 60 Hz. The time proportioning of the relays was controlled by programmable digital counters using a fixed duty

cycle of 5 s. The on or logic high duration of these SSR’s is evaluated by the MPC algorithm. This high logic allows electrical energy to flow through the SSR to the heaters for a determined duration, and therefore, provides the ability to vary the manipulated variables. These parameters were utilized for both the SISO and MIMO controllers. 3.2. Controller parameters Temperature control was based on the following controller parameters: 共1兲 the weighting matrices w and ␦ each assigned as the identity matrix 共no weighting applied兲, 共2兲 a control horizon n u ⫽2 and, 共3兲 a prediction horizon N⫽800. The prediction horizon is based on the time required to reach 95% of the open-loop response and a closed-loop sampling time of 15 s. The tuning variable ␣ was assigned values of 0.0 and 0.5 for the MIMO and SISO control modes, respectively 关see Eq. 共9兲兴. To determine the optimum tuning parameter ␣ that would prevent the control moves from exceeding maximum limits, control simulations were conducted on MIMO and SISO Auto-Regressive Exogenous 共ARX兲 process models given the operating conditions or setpoints, and iteratively adjusting the tuning parameter ␣. This procedure is simple and takes less than 0.01 s to obtain the ␣ values. Matlab System Identification Toolbox was used to formulate these SISO and MIMO ARX process models for these control simulations. 4. Equipment and plastic material The experimental work was conducted on an Engel reciprocating screw injection moulding machine, 34 g shot weight, 222 kN clamping force, and screw length/diameter ratio of 16. Melt temperature measurements for the three heater zones were obtained using a 24 bit analog-to-digital converter with programmable signal conditioning and gain. Timing was facilitated by utilizing 16 bit counter/timer having waveform generation ability. Other peripherals include three solid-state electronic relays with zero crossover. These relays were used to switch the electrical power to the three heating zones on or off. This equipment was used in conjunction with LabWindows 5.01 software that uses functions for data acquisition, advanced mathematical analysis, digital signal processing, and virtual instrumentation. Fig. 4 shows

R. Dubay / ISA Transactions 41 (2002) 81–94

89

Fig. 7. General layout for implementing MIMO MPC.

the electrical schematic of the digital counters relays and data acquisition setup. In the axial direction of the barrel, each thermocouple was placed at the midway of the corresponding zone that was in contact with the molten polymer. The melt temperature probes used are manufactured by Dynisco that can withstand pressures up to a maximum of 2068 bars. The nozzle zone probe was placed a distance of or 0.5 cm within the melt stream and, therefore, measures the true melt temperature. For the other zones, the probes protruded a distance of 1 mm from the inner wall of the barrel. While a true melt reading of the moving plastic due to ‘‘skin effect’’ of the polymer close to the inner wall of the barrel is not obtained, it does provide more accurate measurements of the melt temperature than thermocouples placed in thermowells in the barrel steel wall. The resins used in this investigation were 共i兲 polystyrene 共PS兲 general-purpose grade 678C, and 共ii兲 polycarbonate 共PC兲 CALIBRE 300-15 manufactured by DOW Chemical Company 关24兴. The polystyrene material was used to mould a thin-walled part of weight 20 g and the polycarbonate was used to mould a thin-walled part of weight 32 g. 5. Experiments The experimental tests conducted on the IMM using the control scheme as shown in Fig. 7 are

described, which included open loop testing, the effect of screw rotation and back-pressure, and close loop temperature control for both SISO and MIMO MPC controllers. 5.1. Screw rotation and back-pressure effects The melting of the polymer in injection moulding involves the input of heat energy provided by the zone electrical heaters and other energy sources relating to the injection screw. These other energy sources consist of shear heating by the motion of the screw during mould filling and screw recovery. Investigators 关25–27兴 have reported significant increase in melt temperature on medium tonnage 共greater than 80 tons兲 IMM’s due to viscous friction during translational and rotational motion of the screw, back-pressure during screw recovery, and other moulding parameters as shown in Fig. 1. The effect of viscous heating by the rotating screw was investigated experimentally by allowing steady-state temperatures for all three zones to be attained on the IMM and then rotating the screw at speeds of 100, 200, and 300 revolutions per minute 共RPM兲. The screw was rotated at these speeds for 10 s and during this time with the melt temperature for all three zones measured at discrete intervals of 0.5 s with a shot-stroke of 5 cm. Since rotating at higher speeds results in higher

90

R. Dubay / ISA Transactions 41 (2002) 81–94

Fig. 8. Step input magnitudes to electrical heaters.

throughput, the back-pressure was increased and controlled at values of 600, 1380, and 3500 kPa in order to maintain this fixed screw recovery time of 10 s and retraction distance of 5 cm. Negligible plastic temperature increases of less than 2 °C were measured for all zones during each test. Therefore, the effect of the shear energy and plastication back-pressure on the melt by the rotating screw was considered negligible for this study. This negligible influence can be attributed to the relatively small surface area of the reciprocating screw having a diameter of 25.4 mm. 5.2. Open loop testing and zone temperature interactions The barrel zones and the nozzle zone constitute three process outputs each requiring a control input for control of melt temperature. Therefore, the number of inputs m is 3 and the number of processes to be controlled j is 3. Three step inputs in electrical energy 共the manipulated variable兲 to the heaters were applied individually as shown in Fig. 8 to corresponding heaters. Nine temperature open-loop responses were obtained as in Figs. 9共a兲–共c兲 using a 15-s sampling duration. The three inputs were changed individually by holding the other two inputs constant 共in this case, off兲. Three plastic temperature response curves were obtained to characterize the zone interactions in each case. The three zones were allowed to cool naturally to ambient temperature before the other step input was implemented. These tests were conducted for both an uninsulated and insulated barrel. The discrete response data provided both the SISO and MIMO model for MPC.

Fig. 9. Open loop temperature response showing zone interactions.

5.3. Close loop testing Close loop testing was conducted for both plastic materials with the IMM from a cold start. For the plastic part using polycarbonate 300-15, zones 1, 2, and 3 experienced setpoint changes corresponding to 270, 280, and 290 °C. For the part using general-purpose polystyrene grade 678C, zones 1, 2, and 3 experienced setpoint changes corresponding to 140, 180, and 210 °C. These

R. Dubay / ISA Transactions 41 (2002) 81–94

Table 1 Time instants for setpoint changes on insulated zones. Time 共min兲 0 65 147 207

Insulated zone temperature setpoint °C zone 1 100 140

zone 2 120 180 160 170

zone 3 140 190 200 180

tests were performed on the uninsulated injection barrel. Close loop testing was repeated using the general-purpose polystyrene on the insulated injection barrel using the changes in operating conditions as Table 1. This test was a two-step change in setpoint and was performed to observe the temperature responses using the banded dynamic matrix pb . 6. Results In this section, the results of the open loop tests demonstrating process coupling, and the temperature responses obtained from the practical implementation of MIMO and SISO MPC control for the three melt zones, are presented for typical machine operations. The controller performance was based on obtaining temperature oscillations of less than ⫾1% of the setpoint or operating value, and a settling time within 45 min corresponding to approximately one time constant. The approach taken in this paper was to provide a generic solution for controlling both slow and fast reacting processes by using an unconstrained approach, and a less aggressive setpoint trajectory that initiates from the measured value of each process every sampling instant. The less aggressive setpoint trajectories were chosen by conducting control simulations on SISO and MIMO ARX process models given the operating conditions, and iteratively adjusting the tuning parameter ␣ to ensure that the control actions do not exceed maximum limits. This procedure is very effective and takes less than 0.01 s to obtain these trajectories. The main control program now passes these setpoint profiles 共obtained from the simulations兲 to the routine that controls these processes. It is clear from Figs. 9共a兲–共c兲 that there exist strong interactions between these three zone temperatures when any of the electrical heaters supply energy. The magnitude of the step inputs to each zone as

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shown in Fig. 8 corresponds to two wide bands of operation ( b⫽2 ) of the IMM: 共1兲 a machine idle band and 共2兲 a material processing band during normal operation. From these banded discrete temperature responses, the melt temperature processes are nonlinear as also demonstrated in 关16兴. 6.1. Experimental MPC responses and controller outputs The close loop temperature response comparisons using the MIMO and SISO process models are shown in Figs. 10共a兲–共c兲 for PS and in Figs. 11共a兲–共c兲 for PC using different setpoint operating values. To validate better control performance using the MIMO approach, the desired setpoint trajectories were determined as instantaneous changes from a cold start with the model of Eq. 共9兲 not used or with ␣ ⫽0. From these responses, MPC using the MIMO models provided zeroovershoot responses that significantly reduced the operator setup-time and prevented thermal degradation of the polymer. MPC using the SISO models resulted in some overshoot without oscillations as compared to PID based methods 关2,7,9兴 where the temperature responses were oscillatory. Samples of the MIMO MPC controller output for all zones are shown in Figs. 12共a兲–共c兲 using the MIMO and SISO control modes. Using the MIMO dynamic matrix pb , the controller moves are better optimized when the temperature responses approach the setpoint operating values as compared to the controller moves using the SISO process model. This better optimization is due to the interaction between the zones that are accounted for and modeled in the MIMO MPC scheme. Fig. 13 shows the temperature response for all three zones on the IMM for several setpoint changes that represented normal operating modes, that is, the machine being idle during operator setup, and then the production of plastic components at higher temperatures. The MIMO model was revaluated since insulation was applied to the entire outer surface area of the barrel and electrical heaters. Two dynamic matrices pb were formulated that corresponded to these operating temperature regions. The control algorithm selected the pb matrix for evaluating the predicted profiles b

yˆ r, j and ⌬u. This was achieved by measuring the temperature for all three zones and in software,

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Fig. 10. MIMO and SISO temperature control for polycarbonate part.

Fig. 11. MIMO and SISO temperature control for polystyrene part.

selecting the dynamic matrices pb corresponding to the operating region that the each zone temperature is in. This is the condition for selecting the dynamic matrix pb that is tested every sampling instant. From Fig. 13, the temperature responses showed negligible overshoot of less than 1 °C and good settling time for the required operating conditions as in Table 1. It should be noted that the machine operator was able to conduct these open

loop tests and redefine the predictive controller parameters within an 8-h period without prior knowledge of the control method. The overall approach taken in this study was to provide the machine operator a mechanism for the controller to learn and store its parameters for specific plastic materials, thereby providing better control of the process variables. The control software comprises several routines for 共1兲 conducting

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Fig. 13. Variable setpoint melt temperature response.

therefore exhibit a measure of learning as the plastic material to be processed is changed. 7. Conclusions

Fig. 12. Controller output for MIMO and SISO MPC using polystyrene.

the open loop tests for a specific plastic material, 共2兲 performing system identification using Matlab, 共3兲 conducting control simulations and determining controller parameters using the Matlab models, and 共4兲 passing these controller parameters to a control function that performs the task of regulating the physical processes. The controller parameters, process models, and input magnitudes are all stored automatically on the computer hard disk for the corresponding plastic material and

In this paper, the practical implementation of a MIMO MPC demonstrated good control performance for controlling the melt temperature of three zones on a plastic injection moulding machine. Negligible overshoot of less than 1 °C and fast settling 共within one time constant兲 were achieved when using this approach. PID based methods are not suitable for controlling the melt temperature of the several zones in injection moulding since the entire melting process within the zones is highly coupled and exhibits significant deadtime. Furthermore, the PID control architecture is generally not designed to readily accommodate material and machine variability. This leads to significant temperature oscillations during the transient stage of polymer melting. Melt temperature responses using SISO MPC experienced some overshoot without undershoot, while attaining almost identical settling time when using MIMO control. To minimize the temperature oscillations when using SISO MPC, the modified setpoint profile was used and reevaluated every sampling instant, commencing from the measured value of each process output. This indicates that a less aggressive setpoint trajectory is required for minimizing oscillations when using SISO MPC for controlling polymer temperature. In the case of MIMO control, the modified setpoint profile was not used ( ␣ ⫽0 ) to minimize temperature oscillations implying that a MIMO method for temperature control in injection moulding is desirable and sufficient for tight control. With the thermal inter-

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action between the zones accounted for, the results indicated that less energy is required to maintain melt temperatures thereby resulting in a more energy efficient controller. The self-optimizing MPC structure is better suited for plastic injection moulding since it provides the ability for formulating the dynamic matrix pb for any plastic material, IMM, and operating region. The self-optimizing terminology is based on the ability for the operator of the machine to develop a predictive controller for a specific material without having in-depth knowledge of the control methodology. This technique is a key ingredient for developing controllers on industrial IMM’s that can redefine its parameters when different plastic materials and machine types are used, thereby providing a learning framework for controller adaptability and adjustment.

关9兴

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关14兴 关15兴

Acknowledgments 关16兴

This work was funded by Canadian International Development Agency 共CIDA兲 operating grant held by R.D. The author acknowledges Jack Smith and Dave MacDonald of Ropak Canada Inc. for their technical support and helpful comments, and for the use of Ropak’s industrial machines in testing the control methodology.

关17兴 关18兴 关19兴

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