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System Identification and Multivariable Control of a Distillation Column
SYSTEM IDENTIFICATION AND MULTIVARIABLE CONTROL OF A DISTILLATION COLU¥N T. Takamatsu*, R. Perret**, Y. Naka*, K. Kawachi*
At first, an allowable range of feed composition into a column to make a multivariable(distillate and bottom compositions) control possible is discussed from the point of view of the statics of a column. Next, a method to obtain a simple and low order model for designing a multivariable control system is proposed with the numerical example.
INTRODUCTION In order to design a multivariable control system for chemical processes, at first, the controllability from the point of view of the static characteristics of the process must be examined. In the case of a binary distillation column this examination can be easily done by the graphic consideration with which chem~cal engineers are familiar. After this consideration the multivariable control system must be designed from the dynamic point of view. Generally speaking, the number of state variables in a distillation column is essentially infinite and the number of manipulating variables is finite, and also the strong interactions exist between these variables. Therefore the simple and low order linearized model which is enough to express the characteristics of the interaction must be built. Many researches on the multi variable control of a distillation column have been done (1)~(3) and especially a series of studies on this problem has been reported by Davison (4)~(6). In these researches the method of simplification to the m-th order from the n-th order model, the selection of observed variables, the determination of the values of feedback loop elements, and so on have been considered. In this paper the method of reduction of order by using the concept of system sensitivity is proposed. The method is not based on the simple selection of the eigenvalues with small numerical values, but on the concept of the modification of all eigenvalues such that the calculated values of the output by the model can be best fitted in the values by the higher order model or in the observed values. * Department of Chemical Engineering, Kyoto University, Kyoto, JAPAN ** Department of Control Engineering, Institu~National Polytechniaue, Grenoble, FRANCE
353
Finally the results of digital simulation for the performance of the multivariable control system based on both models by Davison's and our methods are compared with each other. POSSIBILITY OF MuLTIVARIABLE CONTROL FROM THE POINT OF VIEW OF THE STEADY STATE It must be examined whether or not the compositions of distillate and bottoms can be maintained at the desired constant values xd* and xw* for any disturbance of the feed composition zf' As shown by Fig 1 Ca) and Cb), if the practical number of plate s N is larger than the minimum number of plates N which corresponds to the total reflux operation, there is no constraint o~ the disturbance of feed composition for which the conditions of xd* and Xw* can be satisfied by manipulating the rates of reflux and boil up. Of course the flow rates of distillates and bottoms are out of problem. If the condition of the optimum feed plate must be always satisfied, the region of feed composition necessary to maintain xd* and xw* is restricted as shown in Fig.2. An allowable range of feed composition can be determined by obtained the trajectories of both compositionsof the input flow to and the output flow from the feed plate. SYSTEM IDENTIFICATION FOR DESIGNING MULTIVARIABLE CONTROL SYSTEM Assuming that a distillation column can be precisely expressed by an n-th order linear differential l eq~ation and that only m state variables are observable, it may be natural to build the m-th order mathematical model which is perfectly observable and to design a multivariable control system. The n-th order precise model: ~ = Ax + Bu ••••• (1) y = Cx ••••• (2) A; mXn, B; n xl, C; mxn, x; n xl, u; l xl, y; mx l
The m-th order perfectly observable model:
.
x* = A*x* + B*u ••••• (3) y* = C*x* ••••• (4) A*; rnxm, B*; mxl, C*; mxm, x*; mxl, u; lxl, y*; mx l Davison's method for model reduction (4) The basic idea of this method is selecting the m largest eigenvalues from n eigenvalues A P).. 2>···· '>A n of the matrix A and building the m-th order model by the eigenvectors corresponding to).. 1, A 2' •••.. ,Am' The method based on the parameter estimation by using system sensitivity Since Davison's method simply deletes the eigenvalues with small values the response of output given by the m-th order model differs more or less from the response given by the n-th order model. It may be reasonable to modify the values of large eigenvalues in order to reduce the effect of the selection of small eigenvalues. A = [Al A2 J A' rnXrn ••••• (5) A3 A4 ' 1 . Reducing to the m-th order model from the n-th order model is done as follows: (7) A3 A4 = 0, Bl = 0 Putting A2 Al + A*, Bl + B* •••••••• (8) the modifications of are performed.
354
m2 + mxl) , Denoting the parameters included in AI, Bl by Pi(i = 1 " , n 2 - m2 + (n and the parameters included in A2, A3, A4 and B2 by qj(j = 1 , m) xl), the change of the m observable states caused by the parameter variation of Pi and qj can be expressed by 000
000
Ci\x = C(Lj Aji\qj + Li Ilii\Pi)
(9)
00000
where Aj' Il i are the sensitivity coefficients of Aj = dX/aO j and Ili = dX/dPi It is desirable to make difference of y - y* as small as possible, and Eqs. (7) and (8) may be approximately expressed by i\Oj and i\Pi respectively, y - y* =CLj Ajqj - CLi Ilii\Pi = Z - zoo ••
0
(10)
In the above eauation the reduction by Ea. (7) can be done by putting If the reduction is done by using the data R times during the period of dynamic response, the values of 6Pi can be determined by satisfYing Fn( i\ Pi) =
Z=
(~ - Z)T(~ - Z) ~ Min;
(21 ••• zR) T, Z
=
(zl
The solution of this problem is -1 TA 6p = ( 0 T0 ) 0 Zoo o. (12)
(ll)
zR) T 0
~l
,
8 = [C~llT
, 6p =
[6~1
1
eR
CIl R 6Pm(m+d NlmERICAL CALCULATION OF MULTIVARIABLE CONTROL SYSTEM DESIGN Assuming that the n-th order linearized model for a binary distillation column with 9 theoretical plates, condenser and reboiler can be expressed by Table 1. In the design of the multivariable control system based on the concept of Fig.3; the results of numerical calculation applied the values of the control elements to the n-th order system which are obtained by using both the Davison's and the proposed models are shown in Figs.4 and 5. These results are for the case without the feedback loop with Cl in Fig.3. As seen by these results of calculation the control performance based on the proposed model reduction method is better than on the Davison's method from the viewpoints of response time and also steady §tate error. In these calculations, n=ll, m=4, and the method for determining B, ~ and Cp is based on the Davison's method (5),(6). TABLE 1 - A, B of the n-th order linearized model
A=
l -0.174 0.105 0.522 -0.943 0.469 0.522 -0.991 0.529 0.522 -1.051 0.596 0.522 -1.118 0.662 0.522 -1.584 0.718 0.922 -1.640 0.799 0.922 -l. 721 0.901 0.922 -l. 823 l.02l 0.922 -l.943 l.142 0.015 -0.171 B= ~.O 3.28 3.84 4.00 3.76 3.08 2.36 2.88 3.08 3.00 0.3~ x lO-3 ~.O -2.44 -2 . 88 -3.04 -2.80 -2.32 -3.12 -3.82 -4.12 -3.96 -0.4~ Observed variables = xd, x3, x4, Xw
355
DISCUSSION The eigenva1ues of A are ;A
= diag.(-0.030, -0.110, -0.162, -0.264, -0.586, -0.927, -1.321, -1.732, -2.022, -2.681, -3.323)
The eigenva1ues of A* by the Davison's and the proposed methods Al and A2 are Al
diag. (-0.0304, -0.1094, -0.1624, -0.2637)
A2
diag. (-0.0478, -0.1343, -0.2801, -11.321)
It may be seen that all the m eigenva1ues are modified by the proposed method, but are fixed in the Davison's method. REFERENCES ~(9),
1.
Rosenbrock, H.H., 1962, Chem. Eng. Progress,
43.
2.
Gordon-C1ark, M.R., 1964, IEEE Trans. A.C., 1(4), 411.
3.
Gou1d, L.A., 1969, "Chemical Process Control", Addison Wes1ey.
4.
Davison, E.J., 1966, IEEE Trans. A.C., AC-11(1), 93.
5.
Davison, E.J., 1965, "An automatic way of finding optimal control systems for large mu1tivariab1e plants", IFAC Tokyo Symposium on Systems Engineering for Control System Design.
6.
Davison, E.J., 1969, Automatica,
1, 335.
Fig.1 No constraint of feed composition disturbance for the case without the condition of optimum feed plate (a)
, ---I q=l
--
i
-- ---------- --
(b) --- - ---
q=l
feed
i
I I
I
__\j _'V__I
t 'i' _W
356
,--- - - - -- -- --- -_. - -_.
__._._- - -----,--,. Fig.2 An allowable range of feed co~position for multivariable control of a distillation column with the optimum feed plate condition
Fig.3 Construction of multivariable control system X
x*
x=Ax+Bu
o
or
or
x*=A*x*+B*u
w
I i p .4 Comparison of the proposed model to Davison's model reduction for xd
-- ---- ---
Davi son's .-D1Q!ie.l-- - - - - - - -
proposed model - --+ -----+----~
40
30
50
min.
Fig.S Comparison of the proposed model to Davison's model reduction for Xw
xwlmole % 14 r
_
'
12
~
------
_---
~_--:>,
...... ~."...Davison's model
---
,e-
proposed model min.
357
At first, an allowable range of feed composition into a column to make a multivariable(distillate and bottom compositions) control possible is discussed from the point of view of the statics of a column. Next, a method to obtain a simple and low order model for designing a multivariable control system is proposed with the numerical example.
INTRODUCTION In order to design a multivariable control system for chemical processes, at first, the controllability from the point of view of the static characteristics of the process must be examined. In the case of a binary distillation column this examination can be easily done by the graphic consideration with which chem~cal engineers are familiar. After this consideration the multivariable control system must be designed from the dynamic point of view. Generally speaking, the number of state variables in a distillation column is essentially infinite and the number of manipulating variables is finite, and also the strong interactions exist between these variables. Therefore the simple and low order linearized model which is enough to express the characteristics of the interaction must be built. Many researches on the multi variable control of a distillation column have been done (1)~(3) and especially a series of studies on this problem has been reported by Davison (4)~(6). In these researches the method of simplification to the m-th order from the n-th order model, the selection of observed variables, the determination of the values of feedback loop elements, and so on have been considered. In this paper the method of reduction of order by using the concept of system sensitivity is proposed. The method is not based on the simple selection of the eigenvalues with small numerical values, but on the concept of the modification of all eigenvalues such that the calculated values of the output by the model can be best fitted in the values by the higher order model or in the observed values. * Department of Chemical Engineering, Kyoto University, Kyoto, JAPAN ** Department of Control Engineering, Institu~National Polytechniaue, Grenoble, FRANCE
353
Finally the results of digital simulation for the performance of the multivariable control system based on both models by Davison's and our methods are compared with each other. POSSIBILITY OF MuLTIVARIABLE CONTROL FROM THE POINT OF VIEW OF THE STEADY STATE It must be examined whether or not the compositions of distillate and bottoms can be maintained at the desired constant values xd* and xw* for any disturbance of the feed composition zf' As shown by Fig 1 Ca) and Cb), if the practical number of plate s N is larger than the minimum number of plates N which corresponds to the total reflux operation, there is no constraint o~ the disturbance of feed composition for which the conditions of xd* and Xw* can be satisfied by manipulating the rates of reflux and boil up. Of course the flow rates of distillates and bottoms are out of problem. If the condition of the optimum feed plate must be always satisfied, the region of feed composition necessary to maintain xd* and xw* is restricted as shown in Fig.2. An allowable range of feed composition can be determined by obtained the trajectories of both compositionsof the input flow to and the output flow from the feed plate. SYSTEM IDENTIFICATION FOR DESIGNING MULTIVARIABLE CONTROL SYSTEM Assuming that a distillation column can be precisely expressed by an n-th order linear differential l eq~ation and that only m state variables are observable, it may be natural to build the m-th order mathematical model which is perfectly observable and to design a multivariable control system. The n-th order precise model: ~ = Ax + Bu ••••• (1) y = Cx ••••• (2) A; mXn, B; n xl, C; mxn, x; n xl, u; l xl, y; mx l
The m-th order perfectly observable model:
.
x* = A*x* + B*u ••••• (3) y* = C*x* ••••• (4) A*; rnxm, B*; mxl, C*; mxm, x*; mxl, u; lxl, y*; mx l Davison's method for model reduction (4) The basic idea of this method is selecting the m largest eigenvalues from n eigenvalues A P).. 2>···· '>A n of the matrix A and building the m-th order model by the eigenvectors corresponding to).. 1, A 2' •••.. ,Am' The method based on the parameter estimation by using system sensitivity Since Davison's method simply deletes the eigenvalues with small values the response of output given by the m-th order model differs more or less from the response given by the n-th order model. It may be reasonable to modify the values of large eigenvalues in order to reduce the effect of the selection of small eigenvalues. A = [Al A2 J A' rnXrn ••••• (5) A3 A4 ' 1 . Reducing to the m-th order model from the n-th order model is done as follows: (7) A3 A4 = 0, Bl = 0 Putting A2 Al + A*, Bl + B* •••••••• (8) the modifications of are performed.
354
m2 + mxl) , Denoting the parameters included in AI, Bl by Pi(i = 1 " , n 2 - m2 + (n and the parameters included in A2, A3, A4 and B2 by qj(j = 1 , m) xl), the change of the m observable states caused by the parameter variation of Pi and qj can be expressed by 000
000
Ci\x = C(Lj Aji\qj + Li Ilii\Pi)
(9)
00000
where Aj' Il i are the sensitivity coefficients of Aj = dX/aO j and Ili = dX/dPi It is desirable to make difference of y - y* as small as possible, and Eqs. (7) and (8) may be approximately expressed by i\Oj and i\Pi respectively, y - y* =CLj Ajqj - CLi Ilii\Pi = Z - zoo ••
0
(10)
In the above eauation the reduction by Ea. (7) can be done by putting If the reduction is done by using the data R times during the period of dynamic response, the values of 6Pi can be determined by satisfYing Fn( i\ Pi) =
Z=
(~ - Z)T(~ - Z) ~ Min;
(21 ••• zR) T, Z
=
(zl
The solution of this problem is -1 TA 6p = ( 0 T0 ) 0 Zoo o. (12)
(ll)
zR) T 0
~l
,
8 = [C~llT
, 6p =
[6~1
1
eR
CIl R 6Pm(m+d NlmERICAL CALCULATION OF MULTIVARIABLE CONTROL SYSTEM DESIGN Assuming that the n-th order linearized model for a binary distillation column with 9 theoretical plates, condenser and reboiler can be expressed by Table 1. In the design of the multivariable control system based on the concept of Fig.3; the results of numerical calculation applied the values of the control elements to the n-th order system which are obtained by using both the Davison's and the proposed models are shown in Figs.4 and 5. These results are for the case without the feedback loop with Cl in Fig.3. As seen by these results of calculation the control performance based on the proposed model reduction method is better than on the Davison's method from the viewpoints of response time and also steady §tate error. In these calculations, n=ll, m=4, and the method for determining B, ~ and Cp is based on the Davison's method (5),(6). TABLE 1 - A, B of the n-th order linearized model
A=
l -0.174 0.105 0.522 -0.943 0.469 0.522 -0.991 0.529 0.522 -1.051 0.596 0.522 -1.118 0.662 0.522 -1.584 0.718 0.922 -1.640 0.799 0.922 -l. 721 0.901 0.922 -l. 823 l.02l 0.922 -l.943 l.142 0.015 -0.171 B= ~.O 3.28 3.84 4.00 3.76 3.08 2.36 2.88 3.08 3.00 0.3~ x lO-3 ~.O -2.44 -2 . 88 -3.04 -2.80 -2.32 -3.12 -3.82 -4.12 -3.96 -0.4~ Observed variables = xd, x3, x4, Xw
355
DISCUSSION The eigenva1ues of A are ;A
= diag.(-0.030, -0.110, -0.162, -0.264, -0.586, -0.927, -1.321, -1.732, -2.022, -2.681, -3.323)
The eigenva1ues of A* by the Davison's and the proposed methods Al and A2 are Al
diag. (-0.0304, -0.1094, -0.1624, -0.2637)
A2
diag. (-0.0478, -0.1343, -0.2801, -11.321)
It may be seen that all the m eigenva1ues are modified by the proposed method, but are fixed in the Davison's method. REFERENCES ~(9),
1.
Rosenbrock, H.H., 1962, Chem. Eng. Progress,
43.
2.
Gordon-C1ark, M.R., 1964, IEEE Trans. A.C., 1(4), 411.
3.
Gou1d, L.A., 1969, "Chemical Process Control", Addison Wes1ey.
4.
Davison, E.J., 1966, IEEE Trans. A.C., AC-11(1), 93.
5.
Davison, E.J., 1965, "An automatic way of finding optimal control systems for large mu1tivariab1e plants", IFAC Tokyo Symposium on Systems Engineering for Control System Design.
6.
Davison, E.J., 1969, Automatica,
1, 335.
Fig.1 No constraint of feed composition disturbance for the case without the condition of optimum feed plate (a)
, ---I q=l
--
i
-- ---------- --
(b) --- - ---
q=l
feed
i
I I
I
__\j _'V__I
t 'i' _W
356
,--- - - - -- -- --- -_. - -_.
__._._- - -----,--,. Fig.2 An allowable range of feed co~position for multivariable control of a distillation column with the optimum feed plate condition
Fig.3 Construction of multivariable control system X
x*
x=Ax+Bu
o
or
or
x*=A*x*+B*u
w
I i p .4 Comparison of the proposed model to Davison's model reduction for xd
-- ---- ---
Davi son's .-D1Q!ie.l-- - - - - - - -
proposed model - --+ -----+----~
40
30
50
min.
Fig.S Comparison of the proposed model to Davison's model reduction for Xw
xwlmole % 14 r
_
'
12
~
------
_---
~_--:>,
...... ~."...Davison's model
---
,e-
proposed model min.
357