Modelling and Control of a Continuous Frying Process: A Simulation Study

0960±3085/97/$10.00+0.00 q Institution of Chemical Engineers

MODELLING AND CONTROL OF A CONTINUOUS FRYING PROCESS: A SIMULATION STUDY Part II: Control Development L. BRESCIA and R. G. MOREIRA* Friskies R&D Center, St. Joseph, Missouri, USA *Department of Agricultural Engineering, Texas A&M University, Texas, USA

T

he Generalized Predictive Controller developed in this research employs the models mentioned in Part I to calculate future changes in manipulated variables. Physical constraints of the continuous fryer were also imposed on the controller. These restrict the inputs to a certain range regardless of the optimal value. A tracking test was designed using typical industry values. The output setpoints were changed to see how well the inputs tracked the variation. The response time was adequately short. The input variations were within industry standards. The yielded output signals were smooth and showed little or no overshoot for the different setpoint changes. Keywords: GPC; frying; oil content; constraints

INTRODUCTION

Predictive Controller (GPC), has been found to overcome the limitations associated with Minimum Variance and the Smith predictor control schemes6 ,7 . Schonauer8 successfully employed a Generalized Predictive Controller with an ARX model to control a twin screw food extruder. Haarshma 9 applied Dynamic Matrix Control (DMC) to a continuous frying process achieving good results. The objectives of this study were: (1) to develop a GPC controller for a continuous fryer based on the models described in Part I; and (2) to simulate the control performance in closed-loop using different tracking tests.

The Institute of Food Technology has identi® ed the need to improve process design and operation ef® ciencies through closed loop control strategies1 . Process control which maximizes throughput while optimizing product quality and extending the processing time between shut down/clean up would obviously increase ef® ciency. Automatic control of continuous fryers will help to improve ® nal product quality, increase the process ef® ciency, and reduce waste of raw materials. Continuous frying processes are multiple input multiple output systems involving complex interactions2 . These are generally characterized by strong relationships among mass, energy, and momentum transfer including complex physicochemical transformations such as gelatinization of starch, denaturization of proteins, browning reactions, etc. Such changes are in¯ uenced by the chemical composition and physical state of the materials used and by the process conditions. Raw materials used in frying processes are mainly of biological origin and their compositional and physical nature can vary considerably. Such variations can introduce signi® cant, unmeasurable disturbances to the process that make manual control unreliable3 . Besides the complexity caused by raw material variability, continuous fryers also exhibit larger dead time and are typically non-minimum phase systems3 . For systems with signi® cant time delays, improved control performance over PID (proportional integral derivative) controllers has been achieved with the Smith predictor4 and Minimum Variance controller scheme5 providing the time delay is estimated accurately. Incorrect estimates of delays can cause poor performance for these control systems6 . A new class of controllers, Model Predictive Controllers (MPC), is available. One form of MPC, the Generalized

THE METHODOLOGY The fryer, instrumentation, and data acquisition system are described in Part I1 0 . The Generalized Predictive Control (GPC) was used in this study. The controller was adapted for the continuous frying process into Matlab (The Mathworks Inc, MA) code. The process was simulated in closed-loop with a 10% random noise (normal distribution with mean 0.0 and variance 1.0) on the outputs and its performance was tested. Different models developed in Part I were compared. A tracking test was used to test the controller qualitatively. Also, standard deviations were calculated. THE GPC CONTRO LLER Figure 1 illustrates the moving horizon approach of GPC. The basic concept of GPC is, given an appropriately parametrized model and vectors of past plant outputs, future setpoints, previous control inputs, and potential future controls, the predicted output over a range up to the speci® ed prediction horizon, N, can be computed. Increments of control are considered rather than full-valued controls, and beyond a speci® ed control horizon, NU, the 12

MODELLING AND CONTROL OF A CONTINUOUS FRYING PROCESS: A SIMULATION STUDY: PART II 13

Figure 1. Moving horizon approach of model predictive controllers.

control increments are zero. The vector of control inputs is calculated based on optimization of a cost function, usually of quadratic form. GPC involves a receding horizon philosophy where, at each sample time, the free response of the plant is computed, the control increment vector is computed using the optimization routine with future setpoints (with or without constraints), the ® rst control increment is implemented, all vectors are shifted in time, and the procedure repeated at each sample time. The GPC typically uses a time domain stochastic model of the process to calculate future changes in manipulated variables that will minimize a cost function. The GPC adopts an integrator as a natural consequence of its assumption about the basic plant model unlike a majority of designs where integrators are added in an ad hoc way. The ARIMAX (Auto Regressive and Integrated Moving Average with eXogenous input) model form is assumed and successive recursion of the Diophantine equation is used to develop the control equations. The ARIMAX model is of this form: A(q- 1 )y(t) = B(q- 1 )u(t - 1) + C(q- 1 )e(t)/ D (1) 1 where D is the operator (1 - q ) so that D x(t) = x(t) x(t - 1) and A, B, and C are polynomials as de® ned on equation (1)1 0 . The time delay nk (see equation 11 0 ) is absorbed in the B polynomial so that its leading elements are zero. For simplicity, C = 1, resulting in an ARX model. The noise term is assumed to be white. The j-step ahead predictor for such a model is of the form yÃ(t + j/ t) = Gj D u(t + j - 1) + Fj y(t)

(2)

Table 1. Controller parameters values for simulations. Parameter Minimum output horizon (N1) Maximum output horizon (N2) Control horizon (NU)

Value 1 30 25

Trans IChemE, Vol 75, Part C, March 1997

where Gj (q- 1 ) = Ej B, and Ej and Fj are polynomials uniquely de® ned given A and the prediction interval j. They are found via recursion of the Diophantine equations:

= Ej AÄ + q- j Fj 1 =Ej+ 1 AÄ + q-( j+ 1)Fj+ 1 where AÄ = AD . 1

(3) (4)

The vector f can be de® ned as the component of the future plant outputs composed of known terms at time t, so that for example: f (t + 1) = [G1 (q- 1 ) - g10]D u(t) + F1 y(t, and f (t + 2) = q[G2 (q- 1 ) - q- 1 g21 - g20 ]D u(t) + F2 y(t), where Gi (q- 1 ) = gi0 + gi1 q- 1 + . . . Then the predictor becomes: yÃ= GuÄ + f

(5)

where N is the maximum costing horizon, yÃ= [Ã y(t + 1), yÃ(t + 2), . . . , yÃ(t + N)]7 , and uÄ = [D u(t), D u(t + 1), . . . , 7 D u(t + N - 1)] , f = [f (t + 1), f (t + 2), . . . , f (t + N)]. The quadratic cost function: J(N1 , N2 ) = E

+

N2 j=N1 N2 j=1

[y(t + j) - w(t + j)]2

k (j)[D u(t + j - 1)]2

(6)

where N1 is the minimum costing horizon, N2 is the maximum costing horizon, k (j) is a control-weighting sequence, y(t + j) are the future plant outputs, w(t + j ) are Table 2. Control weight sequence for MIMO control simulations. Inputs Oil temperature Submerger speed Takeout speed

Control weight 10 5 1

Outputs

Control weight

colour b oil content

1 1

14

BRESCIA and MOREIRA Table 3. Constraints on inputs.

Input

Low Limit

High Limit

188 50 30

193 75 90

Oil temperature [8 C] Submerger speed [%] Takeout speed [%]

the future set points for the plant outputs, u(t + j - 1)are the controls, is minimized. By manipulating the cost function expectation, the control increment: T D u(t) = gÅ (w - f )

(7)

T (q- 1 ) = Ej AD + q- j Fj

(8)

where gÅ T is the ® rst row of (GT G + k I)- 1 GT , is calculated. For the case of C(q- 1 ) not equal to 1, the controller derivation is expanded. The design polynomial T(q- 1 ) = C(q- 1 ) is introduced. The Diophantine equation (see equation 3) now becomes:

and the predictor turns into: T (q- 1 )Ã y(t + j/ t) = Gj D u(t + j - 1) + Fj y(t)

(9)

or yÃ(t + j/ t) = Gj D uf (t + j - 1) + Fj yf (t)

(10)

1/T(q- 1

f

where is a quantity ® ltered by ). Since the cost function is in terms of D u(t + j) rather than D uf , the prediction equation is modi® ed. Considering: Gj (q- 1 ) = G9j (q- 1 )T(q- 1 ) + q- 1 C j (q- 1 )

(11)

where the coef® cients of G9 and C are found by recursion of the successive Diophantine equations:

= G9j T + q- j C j Gj+ 1 = G9j+ 1 T + q- j- 1 C

Figure 2. Tracking test for X-MIMO controller: (a) step up and down change in output setpoints Ð 6 1.63 in colour b and 6 1.76 in oil content; (b) input responses.

Gj

In practice, a large maximum output horizon (N2 ) is suggested, corresponding to the rise time of the plant. For simple plants, a control horizon (NU) of 1 gives generally acceptable control. Increasing NU makes the control and the corresponding output response more active until a stage is reached where any further increase in NU makes little difference. Table 1 presents the N1 , N2 , and NU values chosen via

(12) (13)

j+ 1

The new predictor in terms of D u becomes

yÃ(t + j/ t) = G9j D u(t + j - 1) + C j D uf (t - 1) + Fj yf (t) (14)

CHOIC E OF CONTROLLER PARAMETERS The choice of output horizons, control horizon, and weighting sequence is critical in controller performance. Clarke et al.6 developed suggestions for selecting output and control horizons. If the plant’ s dead time (k) is exactly known there is no point in setting the minimum output horizon (N1 ) to be less than k. If k is not known or is variable, N1 can be set to 1 with no loss of stability and the degree of B(q- 1 ) increased to encompass all possible values of k. Table 4. Setpoints used in MIMO controllers tracking tests. Setpoint Setpoint 1 Setpoint 2 Setpoint 3

Colour b [ ]

Oil Content [%wb]

61.39 59.76 61.39

23.77 25.53 23.77

Figure 3. Tracking test for ARX-MIMO controller: (a) step up and down change in output setpoints Ð 6 1.63 in colour b and 6 1.76 in oil content; (b) input responses.

Trans IChemE, Vol 75, Part C, March 1997

MODELLING AND CONTROL OF A CONTINUOUS FRYING PROCESS: A SIMULATION STUDY: PART II 15 Table 5. Standard deviations from MIMO controllers tracking tests. Controller

Setpoint Change

Colour b

Oil Content

0.33 0.34 0.33 0.33

0.39 0.39 0.39 0.39

Table 7. Control weight sequence for ARMAX-CB control simulations. Signal

X

from 1 from 2 from 1 from 2

ARX

to to to to

2 3 2 3

simulations. The weights used in the tracking tests, also obtained through simulations, are shown in Table 2. CONTROLLER PERFORMANC E A tracking test was designed to see how well the controller responded to setpoint changes. The performance of the MIMO controllers using the X and the ARX models described in Part I1 0 was compared. Standard deviations from the setpoints were calculated for both simulations. Input variables were forced to be within the ranges speci® ed in Table 3. The product moisture content and oil content setpoints were changed simultaneously in the tracking test utilizing the setpoints in Table 4. The setpoints were changed from setpoint 1 to setpoint 2, i.e., from 61.39 to 59.76 for colour b, and from 23.72 to 25.53% wb for oil content at 500 s. After the process reached equilibrium, setpoints were changed from setpoint 2 to setpoint 3, i.e., from 59.76 to 61.39 for colour b, and from 25.53 to 23.77 % wb for oil content at 3000 s. Figure 2 presents the X controller simulation. Figure 3 presents the ARX controller simulation. Figures 2a and 3a show the simulated process and the setpoint changes for the case of the controller employing the X and ARX models, respectively. Both controllers tracked the setpoints well. Figures 2b and 3b show the input changes that took place in order to reach the new setpoints. To track a +1.63 change in colour b and a - 1.81 change in oil content setpoints, the X controller increased the oil temperature to about 38 C, reduced the submerger speed to about 7% and increased the takeout conveyor speed 25%. Clearly, it can be seen that the takeout conveyor had the greatest effect in the oil absorption of the snack food1 0 . Changes in the inputs for the ARX controller were similar to those of X controller in maginute and directions, with exception to the takeout conveyor speed that was slightly low (about 17%). The same results were obtained for an increase and a decrease in colour b and oil content setpoints, respectively. The standard deviation from the setpoints was calculated to compare controllers performance quantitatively as: n

Sm

=

i =1

Oil temperature Submerger speed Takeout speed Colour b

Control weight 10 5 3 1

where yi is the measured output at sample time i, yspti is the setpoint at sample time i, and n is the number of data points in the segment. Table 5 shows the calculated standard deviations from the setpoints for the two controllers for each setpoint change. The X controller standard deviations are similar to those of the ARX controller for both the moisture content and the oil content. The ARX controller reached the setpoints faster than the X controller. The ARX controller does not show any overshoots when reaching the new setpoints while the X controller shows some overshoot. The controller was also tested using the ARMAX-CB model described in Part I1 0 and the tracking test was performed using the setpoints in Table 6. Input variables were forced to be within the ranges speci® ed in Table 3. The controller parameters used in the tracking test were those in Table 1 and the control weight sequence used is shown in Table 7. Figure 4 shows the simulated process with the setpoint changes, and the input changes that took place to reach the setpoints for the ARMAX-CB controller. For a 6 1.63 change in the colour b setpoint, the controller responded with an increase of 18 C in oil temperature, a decrease of 20% in the submerger speed and no change in the takeout conveyor . The colour b response shows no overshoots. Standard deviations from the setpoints were calculated and are presented in Table 8. In conclusion, for the case of MIMO control, both controllers performed similarly well. In an industrial environment, the ARX controller will be preferred over the X controller because the computational effort is lower

(yi - yspti )2 (15)

n- 1

Table 6. Setpoints used in tracking test of ARMAX-CB controller. Setpoint Setpoint 1 Setpoint 2 Setpoint 3

Colour b [ ] 61.39 59.76 61.39

Trans IChemE, Vol 75, Part C, March 1997

Figure 4. Tracking test for ARMAX-CB controller: (a) step up and down change in output setpoint Ð 6 1.63 in colour b; (b) input responses.

16

BRESCIA and MOREIRA Table 8. Standard deviations from ARMAX-CB controller tracking test. Setpoint Change

Colour b

From 1 to 2 From 2 to 3

0.49 0.51

due to the lower number of parameters in the model. The ARMAX controller performed well also, but its response time was higher (around 1724 s) than those of the MIMO controllers (434 s for ARX and 521 s for X). The standard deviation for the MIMO controllers was 0.39 and for the MISO 0.5. CONCLUSIONS A continuous frying process was analysed in order to model the dynamics and develop a GPC controller. The input variables analysed were the oil temperature, 8 C, the submerger conveyor speed, %, and the takeout conveyor speed, %. The output variables, or product quality attributes, analysed were the colour b, the moisture content, % wb, and the oil content, % wb. The process was modelled using X, ARX, and ARMAX model forms. The developed models showed good results when used in conjunction with a GPC controller. Control weights were chosen as 10, 5, and 1 for the oil temperature, submerger speed, and takeout speed respectively. For the controlled variables, control weights were always chosen to be 1. Values of 1, 30, and 25 were used as the minimum and maximum output horizons, and the control horizon respectively. The MIMO controllers using the X and the ARX models showed similar standard deviations. The ARX controller did not show any overshoot while the X controller showed some. The ARX controller response was a little faster than the one from the X controller. Even though both controllers had similar performances, the ARX controller would be preferred in an industrial environment since it has less parameters to deal with, yielding a lower computational effort. The MISO ARMAX controller with colour b as the controlled variable was also tested. The ARMAX controller did not perform as well as other controllers but gave satisfactory results. Results of this study should be validated on a continuous fryer. The effect of using the PRBS designed in this study instead of the ones provided by the industrial partner should be analysed. Other process variables and product quality attributes besides the ones used here should be studied to ® nd control pairs with better gains. Finally, different control approaches such as ® xed distance and the inclusion of feed forward loops could increase the performance of the control environment.

NOMENCLA TURE ARX ARMAX ARIMAX DMC GPC MIMO MISO MPC N NU N1 N2 PID Sm WR X

auto regressive with exogenous input auto regressive moving average with exogenous input auto regressive integral moving average with exogenous input dynamic matrix control (ler) generalized predictive control (ler) multiple input multiple output multiple input single output model predictive control (ler) prediction horizon control horizon minimum cost horizon maximum cost horizon proportional integral derivative controller performance water rate exogenous input

REFERENCES 1. IFT Research Committee, 1993, America’ s food research needs: into the 21st century, Food Technol, 47(3S): 1S±40S. 2. Moreira, R., 1994, Deep-fat frying of foodsÐ unit operations in food processing, Class Notes (Department of Agricultural Engineering, Texas A&M University). 3. Moreira, R. G. and Sun. X., 1995, Snack Foods: Tortilla chips processing, in Deep Fat Frying, Blumenthal and Porkony (eds) (Chapman & Hall, New York, USA). 4. Seborg, D. E., Edgar, T. F. and Mellichamp, D. A., 1989, Process Dynamics and Control (John Wiley & Sons, New York, USA). 5. Isermann, R., 1991, Digital Control Systems, vol II. (Springer-Verlag, New York, USA). 6. Clarke, D. W., Mohtadi, C. and Tuffs, P. S., 1987a, Generalized predictive controlÐ part I: The basic algorithm, Automatica, 23(2): 137±148. 7. Clarke, D. W., Mohtadi, C. and Tuffs, P. S., 1987b, Generalized predictive controlÐ part II: Extensions and interpretations, Automatica, 23(2): 149±160. 8. Schonauer, S. and Moreira, R.G., 1995, Development of a ® xed-GPC controller for a food extruder based on product quality attributes. Part I: Control development, implementation and analysis. TransIChemE, 73 (C4): 200±210. 9. Haarsma, G., 1994, Development of a dynamic matrix controller for a frying process, Project report (Dept. of Agricultural Engineering and Physics, Wagneningen University, Netherlands). 10. Brescia, L. and Moreira, R.G., 1996, Modelling and control of a continuous frying process: a simulation study. Part I: Dynamic analysis and system identi® cation, TransIChemE, 75 (CI): 3±11.

ACKNOWLEDGEMENTS This research was partially supported by Grants # 999902046 & 999902125 Advanced Technology Program & Advanced Technology Program Development of the Texas Higher Board of Education.

ADDRESS Correspondence concerning this paper should be addressed to Professor R. G. Moreira, Department of Agricultural Engineering, Texas A&M University, College Station, TX-77843, USA. The manuscript was received 12 August 1996 and accepted for publication after revision 20 December 1996.

Trans IChemE, Vol 75, Part C, March 1997