A multivariable self-tuning controller for injection molding machines

107

Applications

A Multivariable Self-Tuning Controller for Injection Molding Machines

C.M. Dong * and A.A. Tseng Department of Mechanical Engineering and Mechanics, Drexel University, Philadelphia, PA 19104, U.S.A.

In engineering process control, a multivariable self tuning controller becomes a requirement when the system dynamics of the controlled process is complex. More flexible than the implicit controllers, the explicit self-tuning controller can be used to provide an effective tool to obtain a variety of high quality products. In this paper, a multivariable self-tuning controller has been proposed for the process control of injection molding machines and its feasibility has been demon- strated by a computer simulation. The present study also suggests that in order to satisfy the requirements for more complicated machines, a learning controller with the additional features of supervision and coordination should be used.

Keywords: Adaptive control system, Self-tuning controller, Multivariables, Injection molding machine, Process control, Computer simulation

Chang Mei Dong is a visiting scholar at the Department of Mechanical En- gineering & Mechanics, Drexel Uni- versity. He graduated in Mechanical Engineering from Harbin Institute of Technology, P.R. China in 1960. He is an Associate Professor of Mechanical Engineering at Kunming Institute of Technology, P.R. China. His current research interests include System Dy- namics, Computer Simulation, Learn- hag and Intelligent Control, AI and Robotics. He is a member of the

Chinese Mechanical Engineering Society and American Society of Mechanical Engineers.

* Visiting Scholar, from the Kunming Institute of Technology, P.R. China.

Elsevier Science Publishers B.V. Computers in Industry 13 (1989) 107-122

0166-3615/89/$3.50 © 1989 Elsevier Science Publishers B.V.

1. Introduction

Since the foundation of self-tuning theory by Peterka [21], Astrom, Wittenmark and Borisson [3,4] and Clarke and Gawthrop [8] in the 1970s, a number of applications of self-tuning regulators (STR) and self-tuning controllers (STC) have shown their effectiveness and efficiency in solving control problems for systems with unknown constant parameters. As an extension to the self-tuning method in the single-input-single-output (SISO) field, multivariable self-tuning theory was developed in various approaches by Borisson [5], Koivo [14], and Koviczky et al. [16,17]. This makes it possible to design controllers for multi- input-multi-output (MIMO) systems in which parameter coupling and interaction are predomi- nant. However, only a few such controllers have been reported in engineering applications.

Among the various MIMO systems in industrial processes, injection molding is a process which is characterized by its remarkable nonlin- earity, dynamic uncertainty and variable interde- pendence. So far, a series of efforts have been

Ampere A. Tseng received his PhD degree in Mechanical Engineering in 1978 from Georgia Institute of Tech- nology. Before joining Drexel Univer- sity as an Associate Professor of Mechanical Engineering in 1985, Dr. Tseng had held various research and development positions in Westing- house Electric Corporation, Martin Marietta Laboratories, and RCA Laboratories. He has written more than fifty technical papers and di- rected various research projects in the

areas of metal forming, computer-aided design and manufacturing, and polymer modeling techniques. Currently he is ac- tive in consultantships for the Aluminum Co. of America, General Electric Co., and Occidental Chemical Corp. He is a member of the American Society of Mechanical Engineers, the Society of Manufacturing Engineers, and the Society of Plastic Engineers.

108 Applications Computers in lndustrv

made to meet the demand for development of an adequate controller for the injection molding processes. Kamal, Patterson and Gomes [13] applied the finite difference method to model the equations of conservation of mass, energy and momentum for each stage of the injection molding process. Their numerical predictions for both the static and dynamic aspects agreed very well with those obtained from the deterministic as well as the stochastic experiments and, in turn, provided a basis to evaluate different control strategies for the melt temperature. Kamal et al. also evaluated different controllers for hydraulic, nozzle and cavity pressures. Based on an empirical approach, Ricketson and Wang [22] developed a process control scheme to obtain high-quality thin plastic products using the injection molding process.

In recent years, various servo control devices and adaptive control systems have been built for injection molding machines to gain flexible control over production and to obtain high-quality products. Costin, Okonski and Ulicny [9] developed a self-tuning regulator to control the injection phase of an injection molding machine cycle. Pandelidis and Agrawal [20] proposed an STR to follow ram velocity profiles for accurate control of the velocity. Nunn and Grolmann [19] suggested an adaptive process control for injection molding. Agrawal, Pandelidis and Pecht [1] reviewed the recent developments in the injection molding process control and identified many research areas in this field, including the stochastic identification, multivariable control schemes, algorithms of determined set points, profiles for various controlled variables, and artificial intelligence applications.

Most of the injection molding research men- tioned above focuses primarily on one processing parameter while cursorily dealing with others. Therefore, it is reasonable to propose research into multivariable adaptive control for overall injection molding systems. In this paper, a multivariable explicit self-tuning controller is proposed, in which all the three fundamental parameters and the two major phases of the overall process of injection molding are controlled as a whole system. The authors' experience in the experiments on a BOY50T2 injection molding machine has been helpful in the research of the controller. The on- line control for the weighting matrices Qa and Q2 of the cost function J2 are found to be effective for obtaining better performance of a MIMO STC.

This has been recommended as a possibly updated approach, i.e. a self-learning controller, to the application of artificial intelligence in process control.

In this paper, after the review of multivariable self-tuning control methods, the stochastic dynamic characteristics of the injection molding process are discussed. An explicit MIMO STC is then proposed based on the system dynamics discussed and the control methods reviewed. A corresponding simulation method is given and a series of simulation results are shown in succession. Fi- nally, a modified self-learning controller is recommended as an improvement of the general MIMO STC on the basis of the present simulation.

2. Self-Tuning Controller for Multivariable Sys- tems

Self-tuning methods deal with control of systems with unknown constant or possibly time- varying problems. This method is mainly employed to improve control precision by reducing the errors caused by the models themselves (differences between the assumptions and the actual conditions) and the variation of system characteristics due to long-term measurement errors. Be- cause the STC adapts to the variation of process parameters, a less precise mathematical model, e.g., low-order linear difference equations, can be used to approach the controlled objects. Certain cost functions are employed in the design of the controller. For a system with unknown parameters, STC can be applied for on-line estimation of the parameters of the controlled system or the controller. The estimated parameters are then fed to the calculator of the controller parameters, which finally is responsible for the self-tuning function of the controller as illustrated in Fig. 1 [12,15,25].

2.1. Explicit Approach

Generally, self-tuning methods consist of two parts: a structure identification algorithm and a control law algorithm. There are two major approaches to the realization of a self-tuning method based on the differences in their organization of the identification algorithm. If the parameters of the system themselves are directly estimated in the

Computers in Industry C.M. Dong, A.A. Tseng // Multioariable Self-Tuning Controller 109

Controll~ paran~r calculation

Controller

Parameter estimation

Controlled [

y(t)

Fig. 1. Schematic diagram of self-tuning controller.

identifier, and the control law is determined by solving a Linear Quadratic Gaussian (LQG) problem, the approach is called explicit self-tuning. On the other hand, when an implicit self-tuning method is used, the estimator parameters are directly estimated by means of the least square method, and then the control law can be determined simply. Although this method has the advantages of reduced calculation and satisfied convergence, it requires the minimum phase of the studied system and an a-priori knowledge of the order and the time delay of the system model. In addition, the restriction on the number of inputs and outputs in a multivariable system and the demand for better prediction for large time delays are also the obstacles blocking the application of the implicit method.

In the case of the explicit method, the need for more time to solve a Riccati equation and the difficulty of convergence are the main problems. However, there are no special restrictions on the number of inputs and outputs and on the minimum phase in an explicit method. Various cost functions can also be adopted when this method is used. These increase the flexibility and potential value of the approach. The crucial factors in achieving on-line application of the explicit method are the simplicity of calculation and the completeness of construction. For the latter, a determination of the order, suborders and time delay of the system model is necessary. Fachs [11] summarized the SISO explicit method in a theo- rem and indicated that it should lead to new and interesting results, especially for nonminimum- phase systems. Salcudean and Belanger [24] discussed the single-variable CARMA model with

the assumption of known model order and time delay.

Since multivariable systems are characterized by the interaction of their parameters and the difficulty of their modeling, manual tuning methods are not easily implemented. Multivariable STCs are, therefore, specially needed for the control of complex dynamic processes. Boyoumi, Wong and Bagoury [6] suggested a multivariable STC for MIMO systems with constant, unknown order and unknown parameters subject to random disturbances and a time delay. Deng and Guo [10] proposed a MIMO explicit STC, which consists of a recursive extended least square (RELS) estimator and a multi-step self-tuning recursive predic- tor-controller with a generalized cost function. Summarizing various MIMO STCs, Toivonen [25] indicated that the explicit method can be applied to systems with unknown time delays and systems that are not stably invertible. It can be expected that as a useful alternative to implicit STC, the explicit method will be studied and applied in more fields.

2.2. Multivariable Explicit Self-Tuning Method

Consider a discrete-time stochastic system described by multivariable linear vector difference equation, ARMAX(n) model [6,10,25]:

A(q-1)y( t ) = B ( q - 1 ) u ( t - d - 1)

+C(q-X)e(t) , (1)

where y(t) is the p-dimensional output vector; u(t) is the s-dimensional input vector; e(t) is the p-dimensional white Gaussian noise sequence of

110 Applications

prediction errors with zero mean; q-1 is the back- ward shift operator, (q -1 )y ( t )=y ( t -1 ) ; and d is the time delay;

A(q-l)=I+ L A i q -i, i= l

?1

B(q-1)=Bo + ~_, Biq-', i = 1

and

C(q - I ) = C o+ ~ Ciq -i. i = 1

Here n is the model order; A~, B~, and C~, are the unknown matrices ( p x p, p x s, and p x p, respectively; and C O = I.

By using the RELS method and a statistical F-test, the parameters A~, B~ and C, and the model order n can be identified. The suborders na, n b and n¢ of the autoregressive, the controlled and the moving-average parts as well as the time delay can be also determined in this way. There- fore, the a-priori knowledge about the order, suborders and time delay of the system model are unnecessary. A reduced model can be obtained as follows:

n a n h

y ( t ) = ~., Aiy(t - i) + ~, Biu(t - i) i = 1 i=d

n,

+ ~., Cie(t - i). (2) i = 0

Assume 0 7 to be the ith row-vector of an unknown parameter matrix

0 = [A, . . . . . A°o; B. . . . . . so~; C~ . . . . . Co~], (3)

i.e.,

0 : 0:] 0;]

Equation (2) can be rewritten into p-subsystems

y , ( t ) = e~(t)0~ + ~ , ( t ) , (5)

where

e ' ( t ) = ( y ~ ( t ) , . . . , S ( t - , ° ) ;

~ ' ( t - 1) . . . . . eT(t - ,~ ) ) .

Computers in lndust O'

At the time t, the RELS estimation of the unknown parameter vector 0~ is

/~(t) = ~ . ( t - 1) + K(t)~,( t ) , (6)

where

~,( t ) = y i ( t ) - ~ T ( t ) ~ ( t - 1) (7)

and

K,(t) = / ' , ( t - 1 ) ,b ( t )

x [z + ~ T ( t ) ~ ' , ( t - a l e ( t ) ] - ' (8)

Here,

(~T(t) = (yT(t-- 1) , . . . , yT(t--no);

u X ( t - d ) . . . . . uX(t--nb);

~T(t -- 1) . . . . . ~T(t _ no)) ' (9)

~ T ( / _ j ) = (~1( t _ j ) . . . . . $p(t --j)) (10)

and

Pi(t) = [I - K,(t)~T(t)] Pi(t - 1)/z. (11)

It is assumed that the forgetting coefficient z = 1 in the present case.

The initial conditions

(o) = O,o, e , (o ) = t",o

are determined by an off-line analysis. Thus we have

~( t )= ~'~(t) , . . . ,Ba(t ) . . . . . Cl(t ) . . . . . ]

~Ti t )

The problem is to find a control taw to mini- mize the following cost function

J = E ~_, D i y ( t + d - i ) - w ( t + d ) i = 0 Q1

+ ~ Giu ( t - 1) , (13) i = 0

where E(.l l t ) , is the conditional expectation; l ( t) = ( y ( t ) , y ( t - 1) . . . . . u( t ) , u ( t - 1) . . . . ); w(t) is the known p x 1 reference sequence; Q1 and Q: are the known p x p and s x s positive definite matrices providing weighting for the cost function

Computers in Industry C.M. Don$ A.A. Tseng / Multioariable Self-Tuning Controller 111

components; D~ and G~ are the known p × p and s x s matrices, respectively; and II x II 2 -- x r a x .

Equation (13) is a generaliTation of the cost function in references [8,14,22]. The cost function includes the two important cases J1 and J2:

J1 = E( [I y( t + d) - y * ( t + d)IIQ,

+llu(t) ll~221It) (14)

and

J2 = E( I1 y ( t + d) - y* ( t + d)I1~,

+ [I u(t) - u(t - 1)11~, I It) • (15)

The self-tuning prediction with k-step recursion can then be obtained from (2) as follows:

na

~9(t+kl t ) = ~ . , . 4 i~9( t+k- i l t ) i= l

n b

+ ~, Biu(t + k - i) i ~ d

n c

+ E C i d ( t + k - i ) , (16) i=k

where k -- 1, 2 . . . . . d. It is assumed that fi(t + k - i l t ) = y ( t + k - i ) for t + k - i < ~ t . If k > n o then the third term of the right side is assumed to be equal to zero. The estimates -4i, J~, Ci and ~(t) are calculated from (6)-(11).

The minimization of J can be proved as the minimization of J * based on the relevance analysis.

d - 1 2Q1

J * = ~_, D i . P ( t + d - i l t ) - w ( t + d ) i = 0

2

+ c o n s t o t i=d Q2

=llDo oU(t) + L(t) I[=o, 2

+ Gou(t)+ ~., Giu(t-1) l +constant, i=1 Q2

(171

where

L( t ) = Do ,~,p(t + d - i l t) i

nb

+ ~., B i u ( t + d - i ) i ~ d + l

"'^ )] + E C i d ( t + a - i

i=d

d - 1

+ E D , p ( t + d - i l t ) - w ( t + d ) , (18) i=1

and if no<d, the third term in the square parentheses is equal to zero; if n b < d + 1, the second term vanishes.

The problem of stochastic optimization here is then changed into a problem of determinate optimization. In this case, let

OJ*/au(t) = 0. (19)

The self tuning controller will be in the form

u(t) = - [ ( DoB d)T Q1DO~d + G~Q2Go ] -1

× L(t)

+GTQ2 ~i=I G iu ( t - 1/] (201

The two important cases are: for the cost function J,,

u(t) = - ( B f Q , Ba+ 02)-IBTQ, L( t ) ; (21)

and for the cost function J2,

u(t)= - (aJQ#,,+ 02)- '

X [B~Q1L(t) - Q2u( t - 1)], (22)

where n a

L ( t ) = ~_, .4i)3(t + d - i l t ) i ~ l

nb

+ ~., B i u ( t + d - 1 ) i = d + l tl c

+ Y'~ C i ~ ( t + d - X ) - w ( t + d ) . (23) i=d

The matrix [(Do~d)TQIDo~d+G~Q2Go] is non- singular for all estimated DoB a matrices. The

112 Applications

matrix (DoBd)TQ1DoBd is symmetric positive semidefinite for D O = 0 and G O = 0, which always yields a positive definite matrix when it is added to the positive definite matrix G~Q2G o. This guarantees a solution for (20) all the time.

The algorithm for the multivariable self-tuning controller is as follows:

Step 1.

Step 2.

Step 3.

Step 4. Step 5.

Step 6.

Determine the order n, suborders n a, rib, nc, and time delay d by means of the multivariable ARMAX model identification; Calculate the estimates -4i, Bi, Ci and ~(t) by using (6)-(11); Calculate the multi-step recursion predic- tor ) ( t + k l t), k = 1, 2 . . . . . d, using (16); Calculate L(t) by using (18); Calculate the self-tuning control u(t) by using (20); and Return to step 2 for the next recursion.

All the above steps are on-line operations except for the off-line step 1. If the parameter estimation in the RELS algorithm is consistent in closed loop, the self-tuning controller (20) will converge to the optimal controller for the corresponding model with known system parameters. The asymptotically optimal (self-tuning) behavior of the controller will be shown in this way.

3. System Analysis

An injection molding is a discontinuous and cycle-repeated manufacturing system of plastic products (Fig. 2). The processed thermoplastic or

cooling or h e a ~ i

clamping / ' . - cylinders /

electric mold heating bands hopper

cylinders

product cavity screw ejector

Fig. 2. Schematic diagram of injection molding machine.

Computers in lndust O,

thermosetting materials are fed into a barrel through a hopper and then sent to the front of the barrel by the rotation of a long screw, which is driven by a hydraulic motor. When heated by the electrical heating bands outside the barrel and the mechanical friction energy inside the barrel, the plastic material melts. Under the pressure of hydraulic cylinders through the screw, high-temperature and high-pressure plastic fluid is injected into the shaped cavity of a mold from a nozzle. It is rapidly spread, cooled, shrunk and solidified in the cavity and finally forms the desired products. After removing the products, the next cycle will be repeated.

3.1. Main Characteristics of Injection Molding Sys- tems

As a thermomechanical and dynamic process, injection molding is first characterized by its mul- tidisciplinary physical phenomena. Rheological data, concepts of energy levels, molecular structure, molecular forces, theory of heat transfer and theory of flow need to be combined to develop a cohesive picture of what happens during the injection molding processes [23]. The exchange of various kinds of energies, mechanical, electrical, ther- mal and fluidal, results in increased complexity of the system dynamics.

With its multi-phase operations during a short time cycle, injection molding essentially does not have a really steady state. Stochastical variations happen frequently in the performance of processed materials, the processing environment, the machine state and technological as well as operational factors. The process dynamic characteristics are changed abruptly at the transfer point between the adjacent phases.

The physical properties of the processed materials, coefficients of heat transfer at the in- terfaces, different modes of heat transfer between heating and cooling, and the behavior of the hydraulic system and dements exhibit all remarkable non linearity [26]. The process parameters in injection molding systems are highly interdependent. Interactions among the molding parameters create new hurdles in the measurement and control of the system. It is obvious from the above attributes that it is extremely difficult, if not impossible, to obtain a complete and accurate mathematical ex- pression for the process.

Computers in Industry C.M. Dong, A.A. Tseng / Multivariable Self-Tuning Controller 113

3.2. Major Process Parameters 4. Control strategy

It can be seen from theoretical analysis and industrial experience that the performance of a multivariable injection molding process mainly depends on three basic technological parameters, i.e. melt temperature, injection velocity and cavity pressure. During the whole process, the melt temperature is a fundamental variable, which has a remarkable influence on the qualilty of the final products. Nozzle pressure, injection rate and cavity pressure are directly affected by the change of the melt temperature. Melt temperature depends predominantly on barrel temperature, but back pressure and screw speed have to be considered t o o .

It is evident that for an injection molding process the variation of injection velocity is closely connected to the internal stresses and molecular orientation of the products. Incorrect injection velocity will result in various defects such as flash- ing, warping and airtrapping. Injection velocity control with servohydraulic technology has been effectively used in plastics manufacturing to provide high-quality products for many years.

In the molding process for plastic materials, the cavity pressure is crucial since it reflects the real state of the desired products inside the mold during the whole process. Injection rate and screw speed are usually employed to control the cavity pressure. A more convenient and effective way, however, is to use the hydraulic pressure in injection cylinders as the controlling agent or parameter, as has been sufficiently proved by industrial experience.

It is noticeable that the three basic parameters are not equally important in the three different phases of the process. Usually, the injection molding process can be divided into three phases: filling, holding and cooling. The system dynamics with respect to the above three basic parameters for each phase are different from the others. In the filling phase, the melt temperature and the injection velocity are the most essential. For the packing phase (holding and cooling), the most effective and representative parameters are the melt temperature and the cavity pressure. The injection molding process, therefore, can actually be considered as a two-parameter multivariable system for each phase.

On the basis of the system analysis and the self-tuning theory, a control strategy will be proposed. A special problem, the transfer method, will also be discussed.

4.1. An Explicit Multioariable Self-Tuning Con- troller

The major goal of process control in manufacturing systems is to gain high productivity and flexibility of the overall process with low variation of the product precision. Handling the demands for a large-range variation of the parameters in the injection molding process is difficult for general PI regulators. This is an obstacle in the improvement of the behavior of the system regulation. Even the determined optimal control does not compensate for the stochastic variation of the process parameters. In this case, a multivariable self-tuning controller is a reasonable choice. The controlled parameters of this system will be continuously updated based on the current process information. Therefore, stochastic disturbances in the process can be attenuated and the variation of the output variables can also be adapted. This will cause the system to achieve a better control target. Based on the above discussion, a two-phase MIMO STC is suited to the injection molding system (Fig. 3).

In the first phase, the melt temperature and the injection velocity will be adopted as the outputs of the system. They are controlled by the barrel temperature and the opening of the servovalve,

e---

L -

Transfer Point

Phase 1: Filling Phase 2: Packing

L / \ __a_~, yl(t): Melt y'l(t): Melt temperature control temperature control

Y2( t): In~-fion y~( t ): Cavity velocity control pressure control

Controlled process

d/_. 1

I I I I I I

Time

Fig. 3. Two-phase multivariable self-tuning control of injection molding process.

114 Appfications

respectively. The outputs for the second phase will be the melt temperature and the cavity pressure, while the barrel temperature and the hydraulic pressure are chosen as the controls (inputs). As for the type of the self-tuning algorithms, although an implicit method is simpler, the flexibility of the explicit method for various conditions is more attractive if the complexity of its algorithms can be simplified. An explicit MIMO STC was developed to control on-line a double effect evaporator [7], the functioning of which was reported to be effective. In fact, there have been some approaches to simplify the explicit algorithms [6,10]. In this paper, an explicit MIMO STC is proposed based on the above principles.

4.2. Transfer Problem

It is necessary to solve the problem of transfer between the two phase-controls. Since the melt temperature is one of the output parameters in both phases, it can be easily maintained in a consistent and continuous state of control. As for the second output parameter, a transfer technique is needed. Actually, only one servohydraulic de- vice is needed to control both the hydraulic flowrate (corresponding to the ram velocity) and the hydraulic pressure in the same injection cylinders. At the transfer point, the control signals of the servovalve opening will be continuously trans- ferred from the measured velocity (the ram velocity) to the measured pressure (the hydraulic pressure). In this paper, the effectiveness of the transfer algorithms has been proved by computer simulation.

The switch-over (transfer) techniques from the first phase to the second based on the ram position, the ram velocity or the cavity pressure were reported to be useful [1,18]. A similar principle will also be helpful for the proposed MIMO STC.

5. Simulation

Computer simulation is a strong tool for testing new control methods and for predicting their effectiveness. In this section, a time-series-analysis model simulation will be performed and the re- suits discussed. A recommendation will be made for the improvement of the present controller.

Computers' in lndustry

5.1. Simulation Method

Consider two second-order autoregressive moving average models with exogenous inputs (ARMAX models) as follows:

For the first phase (filling),

y( t ) = A l y ( t - 1) + A 2 y ( t - 1) + B~u( t - 1)

+ B 2 u ( t - 2) + e(t) + C l e ( t - 1), (24)

where ya(t) is the output of melt temperature; y2(t) is the output of injection velocity; u~(t) is the input of barrel temperature; u2(t) is the input of the servovalve opening; and el(t ) and e2(t ) are white Gaussian noises with zero mean and covariance 02. Here

0 ) [04 0 ] [0.53 0.39 ' 0.51 '

- 0 . 4 6 0.391' B2= 0 - 0 . 4 5 '

[ - 0 . 3 1 0 ] C~ = 0 - 0.46 "

For the second phase (packing),

y ' ( t ) = A~y'(t - 1) + A'2y'(t - 2)

+ B ( u ' ( t - 1) + B 2 u ' ( t - 2) + e ' ( t )

+ C(e'(t - 1),

where y((t) is the output of melt temperature; y~(t) is the output of cavity pressure; Ul(t) is the input of barrel temperature; u~(t) is the input of hydraulic pressure; and el(t) and er(t) are white Gaussian noises with zero mean and covariance e 2. Here

, [0 ] , (0 1 AI__ .26 0 .45 0 0.39 ' A2-- 0.35 '

- 0 . 4 8 0.56J' B2--- 0 -0 .45 '

, [ - 0 . 3 1 0 ] C1 -- 0 - 0.27

In the present simulation, the inputs u(t) and u'(t) are pseudo-random binary series of values +1, while the e(t) and e'(t) are normally distrib- uted N(O, 1) white noise. Based on the above models, the values of outputs and inputs needed for the simulation of the self-tuning algorithms can be directly obtained. In the further simulation, the pseudo random binary series input signals were replaced by the control signals obtained from

Computers in Industry C.M. Dong A.A. Tseng / Multivariable Self-Tuning Controller 115

17.5

'15" ' Y2( t ): Injection velocity

t 2 . 5

~.0.

7 . 5

5. ~ 2.5

Yt( t ): Melt temperature O.

- 2 . 5

- 5 . I . . . . . ! ' I I I I

0 50 I00 £50 200 250 300 Time

Fig. 4. Melt t empera tu re and inject ion velocity as ou tp u t s of the M I M O STC, w = [5, 10] T, Q1 =' I and Q~ = diag[15, 1.5].

the self-tuning controller. The simulation was conducted 300 times on an IBM PC computer.

The input and output values were first used in an off-line algorithm to obtain the order, suborders and time delay for the system model. The order of the system n = 2, the suborders n a = 2, nb= 2, and n c = 1, and the time delay d = 1, for both the two phase models. At the same time, the initial estimate conditions 0io and the initial covariance matrix Pio were also obtained for the simulation of the STC itself. In this case, the key approach is a statistical F-test method. A parsimonious model

with the necessary and fundamental parameters was determined in this way. If a control method is to be tested, it can be seen that the stochastic pseudo-random binary series and white noise signals are as general as the simulation data when compared to the actually measured signals. A multi-step explicit self tuning algorithm proposed by Deng and Guo [10] has been used for the present simulation.

The cost function ./2 in (15) was chosen as the basis of the simulation algorithms so that remarkable effects of the self-tuning control (input)

u:( t ): Servovalve opening

~.o.

ut( t ): Barrel temperature - 2 '

- 4 '

. . . . | 0 • o , o o

--8 0 50 t00 t50 200 250 ' "4

30O

Time

Fig. 5. Barrel t empera tu re and servovalve open ing as i npu t s of the M I M O STC, w = [5, 10] T, Ol = I , and Q2 :ffi diag[15, 1.5].

116 Applications Computers in lndust~

.75

g . r ~

.~ .25"

r~ 0.

- . 25

- . 75"

- "x . o

"~12 ( A l 2 = 0 . 5 3 )

/ f~.23 ( A:051 )

~ (A2o=0.45)

A13 ( A13--0.39 )

'~lo ( AIo =0"36 )

t ) , . , i i i t 50 iO0 150 200 250 300

Time

Fig. 6. Convergence of parameter es t imat ion for -41 and A2-

on the adaptive behavior of the system could be obtained. Different weighting matrices Q1 and Q2 were employed to compare the influences of these matrices on the convergence and control precision of the controller. In the simulation, Q] = / , while Q2 = diag(15, 1.5) and diag(5, 12) for the first and the second phase, respectively. The reference set- tings w = [5, 10]. A step reference series was taken to test the following capability of the controller.

5.2. Simulation Results

The output curves for the two phases are shown in Figs. 4 and 8, respectively. The corresponding input curves can be seen in Figs. 5 and 9. The estimated parameters are given in Figs. 6 and 7 and Fig. 10 for the two phases, respectively. The output and input of the transfer from the first phase to the second are presented in Figs, 11 and

• 75

. 5

• 2 5 '

0 . '

- . 2 5 '

- . 5 '

- . 75"

0

^ B21 ( B 2 | = 0 . 5 8 )

~ Blo ( BI0 =°'51 )

~N~B] 1 ( B11=0.42 )

B~ ( B~a=-0.45 )

~¢~ fi12 ( BI2 ~0.46 )

! i

5 0 i O 0

Bao ( B~=-0.63 )

i 5 0 2 0

Time 2 5 0 3 0 0

Fig. 7. Convergence of parameter es t imat ion for B1 and B2.

Computers in Industry CM. Dong~ A.A, Tseng / Multivariable Self.Tuning Controller 117

t6'

14'

t 2

iO

8

o 8

d

2 y~( t ): Melt temperature

I I I I I I

50 iO0 t.50 200 250 300 Time

Fig. 8. Melt temperature and cavity pressure as outputs of the MIMO STC, w ffi [5, 10] T, Q1 : I , and Q2 = diaglS, 12].

12. The variation of the output and input for a step setting signal are shown in Figs. 13 and 14.

It can be seen from the simulation that the reference setting can be followed by the outputs in the two phases. It is also obvious that the estimated parameters of the RELS are convergent to the corresponding real values except for the parameter matrices C 1 and C~. As the simulation shows, the transfer from the first phase to the second is completed and a reference setting can be reached in a short transient time. The performance of the system when it is affected by a step

input signal is not satisfactory. It seems that a better damping effect for the system of second order models is needed.

As shown in the simulation, all three major process parameters are under effective control of the MIMO STC for the whole process. As op- posed to a single-parameter or a single-phase control, the presented control method is able to real- ize multivariable overall process control approach- hag the main characteristics of the injection molding processes. Although the simulation is only an initial attempt to explore the overall-process self-

4T 2

0 '

- - 2 '

- - 4 '

- - 8 '

- 8 0

~ [( t ): Hydraulic pressure

u j l ( t ): Barrel temperature

| I . , , s

50 t00 t50 200 P.50 300

Time

Fig. 9. Barrel temperature and hydraulic pressure as inputs of the MIMO STC, w ffi [5, 10] T, Q1 ffi 1, and W 2 = diag,5, 12].

118 Appfications Computers in lndust~

.8" //~j B~3 ( BI3=0,56 )

gh (s~1=0.48)

. 4 ' ~ ~

.2" - ~K2o ( A~o=0.45

0 . '

- o 2 '

- . 4

- .6 '

- .8

)

~A'13 ( ,(13 =0.39 ) A23 ( A23=0.35 ) B23 ( B23=-0,45 )

A, Al0 ( A'I0 --'0'26 ) BI2 ( B'12='0'48 )

B20 ( B20=-0.65 )

i t i i | i 50 lO0 i50 200 250 300

Time

Fig. 10. Convergence of parameter estimation for .4~, A~, J~( and B~.

tuning control of a dynamic system in manufacturing processes, and the results are somewhat less precise, it is obvious that the controller has the abilities to overcome stochastic disturbances and to adapt to variations of complex model parameters. This feature is an essential necessity for stochastic dynamic processes such as an injection molding system. On the other hand, it is evident through the simulation that the decoupling of the interdependent parameters and the stability of the

overall system are problems that still need to be investigated.

5. 3. Cost Functions and Weighting Matrices

It is obvious from the simulation that the choice of the cost function is crucial to the performance of a MIMO STC. Between the two cases of the cost function (15) and (16) as well as their corresponding controls (21) and (22), the cost function

t 7 . 5

t 5 . '

t 2 . 5 '

i O . '

7 . 5

2.5'

-2,5'

_ ~ . | i i i i 0 50 t00 150 200 250 300

Time

Fig. 11. Outputs for transfer from phase 1 to phase 2.

Computers in Industry C.M. Dong, A.A. Tseng / Multivariable Self-Tuning Controller 119

51~ ut( t ): Barrel temperature Transfer point

4

! |

-'8 0 50 iO0

u~( t ): Hydraulic pressure

u~( t ): Barrel temperature

t50 200 250 300

Time

Fig. 12. I npu t s for t ransfer f rom phase 1 to phase 2.

J2 and its self-tuning control u are shown to be more controllable. Improvement of the convergence and the control precision for the system can be seen when the cost function J2 is adopted. In fact, the weighting matrix Q2 forms an indepen- dent term with the control u ( t - 1) in (22). This puts an emphasis on the effect of the control u ( t - 1). In other words, the control signals are penalized in a more effective manner in the cost function J2. As Clarke and Gawthrop indicated [8], J2 inserts an extra integrator into the loop and

guarantees equality of y(t) and w(t) at the ex- pense of degraded dynamic performance. There- fore, the controller with the cost function ./2 is more sensitive to the change of the weighting matrix Q2 than that with J1. For the case ,/2, it is easy to regulate the amplitude of the output oscil- lation, and hence also easy to find a suitable weighting matrix for a self-tuning controller.

As for the choice of weighting matrices, Toivonen [25] suggested that the cost function J2 give a convenient parametrization of the optimal

8

i 5

:tO

5

0

- 5

- i 0 '

- i 5

- 2 0

- 2 5 0

Y2( t ): Injection vdodty

YI( t ): Melt temperature

y~( t ): Melt temperature

10 t t t | i iO0 150 200 250 300

Time

Fig. 13. O u t p u t s for s tep sett ing, w = [5, 10] T --~ [ - 5, - 10] T.

120 Appfications Computers in lnd~trr

8 T 8"

2"

O"

-2"

-4"

-8"

- 8 0

t ): Hydraulic prcssur~~'~

i t j t i | 50 t00 t50 200 :=50 300

Time

Fig. 14. Inputs for step setting, w = [5, 10] T---~ [--5, --10] T.

control strategies in terms of the weighting matrices Qa and Q2- QI and Q2 should be chosen properly to achieve the desired closed-loop performance. The proper values for the weights depend on the system dynamics, and in a truly adaptive procedure, the weights should therefore also be adjusted on-line. One approach to adapt the design of the weights is to formulate the control problem as an optimal stochastic control problem with explicit variance restrictions.

In the present simulation, it is found that the choice of the weighting matrix Q2 is critical to the stability of the STC. Very small values of Q2 will lead to the unstable state of the system, and the control signal will also be very large, as indicated in references [6,7]. The performance change of the system is not sensitive to the change of the matrix

Q2 when the values of Q2 are at a high level. It can be concluded from the simulation that an improved performance of the MIMO STC can be obtained by use of the on-line adjustment of the weighting matrices Q1 and Q2.

5.4. A Sel f -Learning Controller

From the above discussion, it is suggested that a self-learning level added to the MIMO STC is expected to result in an intelligent controller with higher performance than that of a general STC. The schematic diagram of this self-learning controller (SLC) is shown in Fig. 15. To obtain this level of controller, it is necessary to update the knowledge from the prior information of the process and the operators' experience. The SLC will

t ~ Supervision & coordination |

Controller ~ Parameter parameter estimation calculation

Cona'olled Controller Process

Self-Learning Level

!

J I y(t)

Fig. 15. Schematic diagram of self-learning controller.

Computers in Industry C.M. Don$ A.A. Tseng / Multivariable Self-Tuning Controller 121

be able to modify the comprehension of the system model by means of heuristically learning and updating the accumulation of the operational data and experience, such as the weighting matrices and forgetting coefficients. The coordination of the heuristic learning and the fundamental self- tuning analysis will be performed through the approach of an artificially intelligent expert control [2]. The SLC also supervises the function of the self-tuning or adaptive level, particularly if the stability and convergence conditions are violated by the operating conditions, for example, if the process parameters change quickly or no external signal excites the parameter estimates [12]. In addition, this controller will be useful in solving the problem of interaction between estimates and control, in using the knowledge base of controlled object characteristics and in taking advantage of the increased hardware proficiency as well as intelligent software research.

is evident through the simulation that the decou- piing of the interdependent parameters and the stability for the whole system are problems that still need to be investigated. A self-learning level with artificially intelligent functions is suggested as a supervisory expert control, to be applied to the self-tuning control.

Acknowledgement

The authors are grateful to Dr. C. Huber and Mr. F.H. Lin for their sincere help in preparation of the paper. We also wish to express our appreci- ation to the Korman Computer Center of Drexel University for permitting the convenient use of its computers.

References

6. Concluding Remarks

In this paper, a two-phase multivariable self- tuning control for injection molding systems is proposed. The physical properties of the materials, heat transfer between heating and cooling, and the behavior of the hydraulic system exhibit remarkable non linearity. The process parameters in injection molding systems are highly interdependent. Interactions among the molding parameters create new hurdles in the measurement and control of the system. Three main process parameters including the melt temperature, injection velocity and cavity pressure are selected to be controlled.

The simulation of this self-tuning control has shown that the controller can provide an effective on-line control for the system considered. The decoupling of the parameters, stability of the overall system and the choice of the weighting matrices need further study. Although the simulation is only an initial attempt to explore the overall-process self-tuning control of a dynamic system in manufacturing processes, and the results are somewhat ambiguous, it is obvious that the controller has the ability to overcome stochastic disturbances and to adapt to the variation of complex model parameters. This feature is an essential necessity for stochastic dynamic processes such as an injection molding system. On the other hand, it

[1] A.R. Agrawal, I.O. Pandelidis and M. Pecht, "Injection molding process control--a review", Polym. Eng. Sci., Vol. 27, No. 18, 1987, pp. 1345-1357.

[2] K.J. Astrom, "Auto-tuning, adaption and expert control", Proc. 1985 American Control Conference, pp. 1514-1519, American Automatic Control Council, 1985.

[3] K.J. Astrom and B. Wittenmark, "On self-tuning regulators", Automatica, Vol. 9, 1973, pp. 185-199.

[4] K.J. Astrom, U. Borisson, L. Ljung and B. Wittenmark, "Theory and application of self-tuning regulator", Auto- matica, Vol. 13, 1977, pp. 457-476.

[5] U. Borisson, "Self-tuning regulators for a class of multivariable systems", Automatica, Vol. 15, 1979, pp. 209-215.

[6] M.M. Boyoumi, K.Y. Wong and M.A. Bagoury, "A self- tuning regulator for multivariable systems", Automatica, Vol. 17, No. 4, 1981, pp. 575-592.

[7] F. Buchholt and M. Kummel, "A multivariable self-tuning regulator to control a double effect evaporator", Auto- matica, Vol. 17, No. 5, 1981, pp. 737-743.

[8] D.W. Clarke and B.A. Gawthrop, "Self-tuning controller", Proc. lEE, Vol. 122, 1975, pp. 929-934.

[9] W.H. Costin, D.A. Okonski and J.C. Ulicny, "Control of an injection molding machine: adaptive regulation during filling", 1987 American Control Conf., American Auto- matic Control Council, 1987, pp. 711-716.

[10] Z.L. Deng and Y.X. Guo, "A multivariable explicit self- tuning controller", lnt. Conf. Control 85, lEE Conf. Publ. No. 252, University of Cambridge, 1985, pp. 23-27.

[11] J. Fachs, "Explicit self-tuning methods", IEE Proe., Vol. 127, Part D, No. 6, 1980, pp. 259-264.

[12] R. Isermarm, Digital Control Systems, Springer-Verlag, Berlin, 1981.

[13] M.R. Kamal, W.I. Patterson and V.G. Gomes, An injection molding study, Part I, Melt and barrel temperature dynamics, Polym. Eng. Sci., Vol. 26, No. 12, 1986, pp. 854-896.

122 Applications Computers in Industry

[14] H.N. Koivo, "A multivariable self-tuning controller", Au- tomatica, Vol. 16, 1980, pp. 351-366.

[15] Y. Koren, Computer Control of Manufacturing Systems, McGraw-Hill, New York, 1983.

[16] L. Koviczky and J. Hetthessy, "Self-tuning minimum variance control of MIMO discrete time systems", Auto- mat. Control Theory AppL, Vol. 5, No. 1, 1977, pp. 11-17.

[17] L. Koviczky and K.S.P. Kumar, "Multivariable self-tuning regulator with generalized cost function", lnt. J. Con- trol, Vol. 33, No. 5, 1981, pp. 913-921.

[18] R.A. Malloy, S.J. Chen and S.A. Orroth, "A study of injection to holding pressure switch-over techniques based on time, position and pressure", Conf. Proc. 45th Annu. Tech. Conf. & Exhib., Society of Plastics Engineers, 1987, pp. 225-230.

[19] R. Nunn and C. Grolman, "Adaptive process control for injection molding", Conf. Proc. 46th Annu. ,Tech. Conf. & Exhib., Society of Plastics Engineers, 1988, pp. 298-304.

[20] I.O. Pandetidis and A.R. Agrawal, "Self-tuning control of ram velocity in injection molding", Conf. Proc. 45th Annu. Tech. Conf. & Exhib,, Society of Plastics Engineers, 1987, pp. 235-257.

[21] V. Peterka, "Adaptive digital regulation of noisy systems",

2nd IFAC Syrup. Identification and System Parameter Estimation, International Federation of Automatic Con- trol, Prague, 1970.

[22] R.C. Ricketson and K.K. Wang, "Injection molding process control based on empirical models", Conf. Proc. 45th Annu. Tech. Conf. & Exhib., Society of Plastics Engineers, 1987, pp. 298-304.

[23] I. Rubin, Injection Molding: Theory and Practice, Wiley, New York, 1973.

[24] S. Salcudean and P.R. Belanger, "Adaptive regulator: computational algorithms and simulation results", 6th IFA C Syrup. Identification and System Parameter Estima- tion, Washington, DC, International Federation of Auto- matic Control, 1982, pp. 198-203.

[25] H.T. Toivonen, "Techniques for multivariable self-tuning control", Control and Dynamic Systems, Advances in The- ory and Applications, System Identification and Adaptive Control, Vol. 26, Part 2 of 3, Academic Press, New York, 1987, pp. 189-224.

[26] A.A. Tseng, P.F. Sun and S.P. Koiluri, "Computer-aided modeling of polymer processing: a review", Conf. Proc. ANTEC "89, Society of Plastics Engineers, 1989, pp. 267-270.

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A multivariable self-tuning controller for injection molding machines