A Comparative Study of Linear and Nonlinear Multivariable Binary Distillation Column Control

Copyright © IF AC Dynamics and Control of Chemical Reactors (DYCORD+'92), Maryland, USA,1992

A COMPARATIVE STUDY OF LINEAR AND NONLINEAR MULTIV ARIABLE BINARY DISTILLATION COLUMN CONTROL R. Pullinen, P. Pietila, T. Jussila and P. Lautala Control Engineering Laboratory, Tampere University o/Technology, P.O . BoT. 527, SF·33JOJ Tampere, Finland

Abstract. Distillation columns present several problems for a control engineer because of their nonlinear nature and strong interactions. In this paper model predictive control and multivariable PI control are compared. Also, a method for constructing a nonlinear process model and linearizing it using Simulab software is presented. The controller simulation results indicate that model predictive control behaves better than PI control. Anyhow the ease of tuning of the PI controller makes it a very attractive choice.

Keywords.

Process models, Predictive control, Linear control, Distillation columns.

THE PROCESS

THE BILINEARMODEL ANDLINEARIZATION

The process used for experiments presented in this paper is a simple two-product distillation column, Fig. l. At steady state the column has two degrees of freedom which may be used for example to determine the top and bottom compositions (Yn and XB). From the control point of view, the controller of this 5 x 5 system should manipulate all five inputs (L, V, VT , D, B) in order to keep the five outputs (top and bottom level, pressure, top and bottom composition) as close as possible to desired values. The controller should use all available data: process measurements, model and expected disturbances. In practice decentralized control systems with five single control loops are used instead of full 5 x 5 controller.

The bilinear model of the column is based on the temperature balances between successive trays. The balance in the top tray is

The balance in the trays (T2 - T6) is

~dT = (L * (T i. l - n +

C;

* (V+Vl) * (T i+l-Tj)) IM (2)

t

The balance in the feed tray (17) is dT7 = (L*(T6 - T7l+C7*(V +VI )*( Tg-T7l+F*(Tr - T7)) I M

dt

(3)

The balance in the lower trays (T8 - T 12) of the column is

The binary liquid used in the process is ethanol-water. The total height of the column is 4 meters, diameter is 30 cm and there are 12 trays. Because of the size of the column, there are some physical restrictions. First, the maximum temperature difference allowed between bottom and top is 8°C, otherwise the column is uncontrollable. Second, the pressure differences inside the column are neglible because of the size of the column , so the pressure is not controlled. There is a strong interaction from column bottom to top. There is also a strong disturbance that cannot be predicted from heat exchanger LS 1 to feed flow temperature affecting the column temperature balance. The selected control strategy is of conventional two point LV configuration.

-idT ·

=(L + F) * (Tj-I - Tj) + Cj * (V+ VI) * (Tj+I-Tj)) I M

(4) where Ti is the tray in question (referring to Fig. I), R is the external reflux (kg/sec), Tr is the temperature of the external reflux (0C), L is the internal reflux, Ci is the ratio of Cv/Cl for the tray in question, V is the vapour fl ow in the column, VI is the vapour flow change due to the change in steam valve position from its base point (%), M is the mass units per tray of binary liquid, F is the input flow and Tf is the temperature of the input flow (0C). Three of the lower trays (T9 - Tll) have been combined as one tray in the model for simplicity.

231

The heat loss caused by the temperature of the feed flow has been taken into account by adding term Q6 to the equation of tray T6. Two models were constructed. one without delays and one with delays included. Delays are included for all trays from both inputs. the reflux flow and reboiler steam flow.

0.0003 0.006 0.001 0.008 0.001 0.0002 0.0003 -0.004 0.0001 0.000 1

B=

The model parameters in steady state are: R =0.00771 M = 1.15000 Q6 = 0.12831 C3 = 0.529 C6 = 6.02 Cls=0.609

L=0.00830 F= 0.()4625 Cl= 4.373 C4= 0.261 C7 = 1.00 C12= 1.853

V =0.024456 Tr= 52.7 C2=0.677 C5= 0.931 C8= 0.146

D=

[~

~J

(8) (9)

A software package for nonlinear model predictive control has been developed by Golemanov et al (1990). The algorithm is intended to be used to solve control problems with restrictions and delays. Further developement with the algorithm has been done by Golemanov et al (1991). The model used in the package is described with discrete state equation

The linearization of the model is done in three operating points. the first of which is the basic operating operating point of the column with reflux 30 IIh. boiler steam flow 30 % and feed flow 180 IIh . The second operating poin ts differs from the first only with boiler steam flow value of 35 % and the third operating point with the reflux value of 40 lIh. The linearization was done using SIMULAB's linmod command which produces the linear model in normal state space form:

qx (k) =A x (k) + B h (TD) uv(k) =C x (k) + D u (k)

y (k)

(10)

where = t-Kh •.. .• t+Kq-l. = the amount of sampling intervals in measurement horizon Kq = the amount of sampling intervals in prediction horizon q = prediction operator. h = delay operator. x = state vector u = control vector uv = vector for combined control and modelized disturbances y = output vector A. B. C. D = system matrices and TD = delay matrix of control signals and modelized disturbances k Kh

(5)

When the data is in state space form. the function's of MATLAB's Control System Toolbox can be used for further analysis and the functions of Control System Toolbox and Robust Control Toolbox for linear control system design. In basic operating point the linearized model without delays is A= 0 0 0.007 0.01 1 0.046 0 0 0 -0.020 0 0 0 0 -0. 100 0.093 0 0 0.007 -0.022 0 0 0 -0.013 0 0 0 0 0.007 0 0 0 0 0 0 0 0 0.00 3 0 0 0

[100 0 0000 00J 01000 0 0 00 0

MODEL PREDICTIVE CONTROLLER

The actual models were built with SIMULAB. a program for simulating dynamic systems. It is an extension to widely used MATLAB. adding many features spesific to dynamic non linear systems. The SIMULAB model of the column is shown in Fig. 2 and the model of one of the trays. T3 in Fig. 3. Both models were also coded with SIMNON for comparison.

·0.019 0 0 ·0.085 0.004 0 0 0 0.014 0 0.007 0 0 0 0 0 0 0 0 0

C=

(7)

All models are non minimum phase.

where Cls is the ratio of Cv/Cl for combined trays (T9-Tll).

:x:=Ax + Bu y=Cx + Du

-0.839 4.584 0.433 -21.586 -1.674 -0.538 -0.699 -22 .514 0.992 0.0 88

0 0 0 0 0 0 0 0 0 0 0 0.0 15 0 0 0 0 0 0 0 0 0.005 0 0 0 ·0.027 0.020 0 0 0.007 -0. 135 0. 128 0 0 0.007 -0 .029 0.0 21 0 0.046 -0.049 0

The behaviour of the controller can be changed with the parameters of system model affecting the prediction of outputs. the parameters affecting the modelization happening before present time and the sampling frequency. When selecting these parameters following has to be taken into account: - the prediction horizon TQ must be equal or longer than the shortest time constant of the process TC min • preferably TQ = 3 TCmin. - measurement and prediction horizons must be longer than the longest delay in the process TDmax + Dtx• where Dtx is the sampling frequency of measurement and prediction horizons. respectively.

(6)

232

If these conditions are met, the controller can take into account the effects happening in the process dynamics after the delay.

was tuned for the system. The model used for the tuning was the linearized model of column in basic operating point with delays included. Kp and K, are the gain matrices of proportional and integral parts of the controller respectively. The task of a tuning method is determine these two matrices based on some kind of process model or experimental measurements. In this case a tuning following the ideas of Penttinen and Koi vo (1980) is used. The tuning is based on the process models introduced before. One of the tuning proposals of Penttinen and Koivo for stable systems of the form

The algorithm offers versatile possibilities for using system spesific restrictions: - each control signal u can have minimum and maximum value for each part of prediction horizon - boundaries can be set to the value of control signal change happening at any time in prediction horizon - upper and lower limits can be set to the values of controller state x - the predicted output yp can be restricted between suitable limits - the control signal has internal restrictions

Xi (t) = yet)

L IYj (k) - Ymj (k) I+ phj IYj (t-I) - Ymj (t-I) I+ k = t-Kh

L Irj (t) - Ypj (tll + pqj eaj

(11)

where = measured value

= modelized value = predicted value =set point = additive error = coefficients

subspace of the positive (c: , 8) plane was determined using the MFD Toolbox (Ford, Maciejowski, and Boyle, 1988) together with self-made Matlab functions. Then a few trial tunings were picked from this stability plot. The final choice was c: = 0.002 , 8 = 0.005. The tuning was

PI CONTROLLER

=

u(t)

e(t)

= Kp e(t) + K, vet)

Kp

= 0.002

[717.4 131.3] -0.9354 0.0470

K,

= 0.005

-2.387 17040] [ -0.00 I 0.0024

Some simulation results of set point step changes are presented in Fig. 6 and Fig. 7.

For comparison, a multi variable PI controller of the form - yet)

(17)

The positive numbers c: and 8 are the fine tuning parameters of the controller. They are chosen small enough, usually in the interval (0,1). A "stabilizing"

The controller used with the distillation column of Fig. 1 controls the top temperature T3 and the bottom temperature Tll. The disturbance is the change in feed flow FIC 11. Some simulation results of set point step changes are presented in Fig. 4 and Fig. 5.

vet)

(16)

= C xCt)

( 18)

Summing all cost functions the total cost function J is formed in accordance to which the control trajectory is optimized. The optimizing problem defined by J and the restriction equations is solved with linear programming (MIMC Operating Manual, 1991). The result is a set of control signals covering the prediction horizon, only the first of which is sent to the system.

yre~t)

(15)

if the required inverse exists. This selection aims at, decoupling the system at high frequencies (Maciejowski 1989, p.142).The rough tuning matrix Kp can also be determined graphically from the open-loop step responses of the system. The rough tuning matrix for the I-part of the controller is selected as

k=t

e(t) =

Bij Uj (t-d ij )

j= 1

Kp= c:(CBr 1

t + Kq-1

Yj Ymj ypj

L

is that the proportional part of the controller is selected as

t -2

pqj

m

Aij Xj (t) +

j= 1

The cost function is J j = phj

L

(12) (13)

(14)

233

COMPARISON OF CONTROLLERS

REFERENCES

The main benefit of using model predictive control algorithm is its ability to effectively control systems with delays and to take into account control signal restrictions internally without using the classic and not very efficient way of checking restrictions only after the control signal has been calculated. The main weakness of the algorithm is the diccifulty of tuning: the connection between parameter selection of the target function and the control results is not unambigious. Another weakness of the algorithm was found when experimentary control runs were made with the binary distillation column of Fig. I. When the process controlled is very non linear and the linearization of the model is done in very narrow area around the desired operating point the behaviour of the controller is very heavily dependent on the accuracy of model parameters.With a bad model the tuning parameters do not affect the control result very efficiently.

Ford, M. P. , J. M. Maciejowski and J. M. Boyle (1988): Multivariable Frequency Domain Toolbox : Users Guide. Cambridge Control Ltd. Golemanov, L. A., V.Atanassov, Z. Banchevsky and A. VeIkov (1990): A Model Incorporating Multifunctional controller (MIMC). Report 83, Helsinki University of Technology, Control Engineering Laboratory, Finland. Golemanov, L. A., P. Lautala, V.Atanassov, Z. Banchevsky, G. Elenkov, P. Pietilii, R. Pullinen and A. Velkov (1991): A Semi-industrial Implementation of Model Incorporating Multifunctional controller MIMe. Report 6/91, Tampere University of Technology, Control Engineering Laboratory, Finland. Maciejowski, J.M. (1989). Multivariable Feedback Design, Addison-Wesley, Wokingham, England.

Another benefit of the model predictive algorithm studied here is its ability to run the process in several operating points. One of the aims of the development team was to develop an algorithm which is also able to do process startup sequences. This, however may b.e somewhat ambitious demand, since at startup all varIables are changing at the same time, that is, the process state itself, the model used and all control signals. A better possibility to do startup sequences might be to use some other algorithm, for example fuzzy control for startup and when the process is in its operating region switch to model predictive controller.

A Model Incorporating Multifunctional Controller (MIMC). Operating Manual, ver. 1.0. (1991). Helsinki University of Technology, Control Engineering Laboratory, Finland. Penttinen, J. and H. N. Koivo (1980). Multivariable tuning regulators for unknown systems, Automatica, Vol. 16, No. 4, pp. 393-398.

The PI controller simulations show that PI controller is not able to take care of the interactions as efficiently as the model predicti ve controller. Response times are also clearly longer with larger overshoot in column bottom. Since in real process industry often the quality of only one of the products of the column is important, the behaviour of the PI controller could be improved by tuning the main product control loop "tighter". This, however gives worse results with the other loops. The main benefit of the PI controller compared to model predictive algorithm is its ease of tuning. The change of tuning parameters unambigiously shows in the process response and with aid of suitable software the tuning and simulation experiments can be done reasonably easy before using the controller with the real process.

CONCLUSION In this paper a model predictive control algorithm was compared with multi variable PI controller. The results suggest that although the operation of model predictive controller is better, the PI controller is much easier to tune and clearly simpler to construct.

234

Fig.I. The binary distillation column of Tampere University of Technology with basic instrumentation.

input DV

VI

Mu.

Fig. 2.

The SIMULAB model of the column 235

Fig. 3.

The model of tray T3

12

12

0.8

0.8

0.6

0.6

0.4

0.4

0.2

0.2

o -0.20!:-------:5~00:------:-17:00:-:-0--....,1~500::-:---2::-:07.00:---~2500

Fig. 4.

-0.2

o

Fig. 5.

Model predictive controller. Step response of column top temperature with interaction to bottom temperature

1.5 r----~---~---~--____,

'500

1000

1500

2000

2500

Model predictive controller_ Step response of column bottom temperature with interaction to top temperature

1.5r---~--~------~---,

,.....-/~.~.............-.-:::~-:::::::::-:,:::>=-.,.~--­

,---- -- - -----=---::::---;:::;-=-= -- ; = = - - -

I

I

I '

0.5

0.5

1/ r

f

if

1

Or----"'1 ...,."...- - - -_

i

Or----~

............. .

-0.5 ' - - - - - - - - - - - - - - - - - - - ' o 1000 2000 3000 4000 5000

-0 . 5L------------~---...J

o

Fig. 6.

1000

2000

3000

4000

PI controller. Step response of column top temperature with interaction to bottom temperature

Fig. 7.

236

PI controller. Step response of col umn bottom temperature with interaction to top temperature