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Nonlinear control of an industrial reactive distillation column
IFAC
Copyright !Cl IF AC Dynamics and Control of Process Systems, Jejudo Island, Korea, 200 I
C:UC> Publications www.elsevier.comllocate/ifac
NONLINEAR CONTROL OF AN INDUSTRIAL REACTIVE DISTILLATION COLUMN S. Griiner * K.-D. Mohl * A. Kienle ** E. D. Gilles ., •• G. Fernholz"· M. Friedrich *** • Universitat Stuttgart, Institut fUr Systemdynamik und Regelungstchnik, Pfaffenwaldring g, 70550 Stuttgart, Germany . .. Max-Planck-Institut fUr Dynamik Komplexer Technischer Systeme Magdebury, Zenit-Gebaude, Leipziger Strafle 44, 39120 Magdebury, Germany **. Bayer A G, Process Technology, Process Control, 51368 Leverkusen, Germany
Abstract: Control of reactive distillation columns is a challenging task due to the complex dynamics arising from the coupling of reaction and separation. In this paper asymptotically exact input/output-linearization is applied to an industrial reactive distillation column. This control scheme requires knowledge of the complete state of the process and therefore an observer is designed. To compensate for steady state observer offsets an outer control loop with simple PI-controllers is implemented. In comparison to a well tuned linear controller the nonlinear controller shows a superior performance with respect to setpoint-changes and disturbances, even in the presence of unknown input delays. Copyright © 2001lFAC Keywords: nonlinear control, observers, robustness, reactive distillation columns
active distillation column, including the observer design, based on simple temperature measurements is considered. After a brief description of the process model the basic idea of the control concept, namely input/output-linearization is presented and applied to the process. Then a suitable observer is designed and tested in simulation studies. Finally the robust performance of the nonlinear controller is compared to that of a well tuned linear controller (Fernholz et al., 2001) wrt. set point changes and disturbance rejection.
1. INTRODUCTION
Reactive distillation has recieved a tremendous industrial interest over the last years. Many publications have been made regarding the design of such columns, however concentrating on the steady state behaviour. Even more striking are the complex dynamics of reactive distillation columns, due to the coupling of reaction and separation. Nevertheless only few publications are available regarding the control of reactive distillation columns and most of them consider only linear control e.g. (Sneesby et al., 1999). Those employing nonlinear controllers assume that the state of the plant is completly measurable and thereby neglect the need to design an observer that performs well in the closed loop e.g. (Kumar and Daoutidis, 1999). Others rely on concentration measurements that are rather difficult to obtain on a real plant e.g. (Monroy-Loperena et al., 1999) .
The process under consideration is an industrial reactive distillation column schematically shown in Fig. 1. The column consists of NS = 63 trays including the reboiler as tray NS and the condenser as tray 1.
In this contribution, to our knowledge for the first time, nonlinear control of an industrial re-
The feed enters on tray kF = 25 and contains the components Band C from which the product E
2. PROCESS
113
Ne
0= 1- LYi,k
top key component: C
(5)
;=1
B.CL F ~ . ", liquid reed f-~-
A+E"'B+D
.
:
holds. The pressure is computed according to 0= Pk - D..p - Pk-l , PI = const. (6) and the model is completed by the summation condition for the liquid-phase mole fractions
Cbemical Reactioas:
A+D"'B+C
~-T.\
Ne 0= 1- LXi,k. i=1
k k+1
The main nonlinearities are introduced by the thermodynamic quantities. Namely the activity coefficients li,k, computed according to Wilson and the saturation pressures Ps,i, computed according to Antoine and a polynomial approach depending on the component (Ried et al., 1987). Another non linearity are the reaction rates rO,j,k which are based on an Arrhenius-like formulation.
bonom product E bonom key component: 0
Q
Fig. 1. Schematic view of the column is produced. In the fluid phase NR = 2 reactions with reaction rates rO,j,k occur and the simplified reaction scheme is given in Fig. 1. The lower boiling component A is removed as the top product and its purity is specified in terms of a threshold for the top key component, the impurity C. The main product E is removed form the column as bottom product and its purity is specified in terms of thresholds for the bottom key component, the impurity D.
For the reboiler (k = NS) the balances (1)-(3) are adjusted since there is no VNS+1 and the energy balance (3) is extended by the additive term Q representing the reboiler heat input. The material balances (1) and (2) are replaced for the condenser (k = 1) by equations stating no composition change from tray k = 2 and equal liquid and fluid composition. The energy balance (3) is dropped and replaced by the equation for the reflux ratio R = Lt/VI'
Previous results reported in (Groebel et al., 1995) for non reactive distillation indicate that there is no need to explicitly include fluid dynamics in t.he design model. Hence the model is based on the assumption of constant volume holdup and therefore the molar holdup nk depends on the composition. For each of the ordinary trays k = 2(1)NS -1 the model consists of the dynamic component material balances dx· k nk d~'
= Lk-lX;,k-l -
The plant model consists of NS· (NC -1) dynamic and NS· (NC + 5) algebraic equations corresponding to 252 dynamic and 630 algebraic states. 3. BASIC IDEA OF INPUT /OUTPUT-LINEARIZATION
LkX;,k
NR
Instead of a detailed presentation of the underlying theory only a brief sketch of the concept is given and emphasis is on the application. For details on the theory the reader is refered to (Isidori, 1995; Nijmeijer and van der Schaft, 1990) or (Friedrich et al., 1992) . An application to nonreactive distillation can be found in (Groebel, 1996; Groebel et al., 1995).
xn '
F +Vk+1Yi,k+l- VkYi,k+ LVolkvi,jrO,j,k+(L j=l i = l(l)NC - 1.
(1)
The flow rates are computed from the stationary total material balance
The system under consideration can be represented by
0= Lk-l - Lk + Vk+1 - Vk + (LF) Ne NR L Volkvi,jrO,j,k
+L
(2)
x = f(x, z, u),
;=1 j=l
0= k(x, z, u),
and the stationary energy balance
+ Vk+lhV,k+l VkhV,k + (LFh F) .
w
0= Lk-lhL,k-l - LkhL,k -
= q(x,z)
(8) (9) (10)
for t > 0 and the initial conditions x(t = 0) = Xo together with 0 = k(Xo,zo,uo), x E JRn,z E JRI and u, W E JRm i.e. a quadratic multi-input/ multi-output (mimo) system.
(3)
Since standard enthalpies of formation are used, no reaction term appears in (3) . In eq. (1)-(3) the terms in brackets appear only for the feed tray. Besides the vapour-liquid equilibrium relation Ps,i O = Y; ,k - - , i,kXi,k Pk
(7)
3.1 Derivation of control law
(4)
The idea of input/output-linearization is to differentiate the elements Wl wrt. time t, replacing all
the boiling point condition
114
terms x by f(x, z, u) until an expression explicitly involving at least one component of u is obtained. The number of differentiations needed is called relative degree T and for mimo systems the vector r = h, T2 , ... , TmJ is called relative vector degree.
to linearize the observer dynamics instead of the plant dynamics and to interpret the error injection of the observer as a disturbance. Due to the asymptotic convergence of the observer to the plant an asymptotically exact input/outputlinearization is obtained.
For systems with T/ = 1 and q/ depending only on the differential variables x this procedure yields . = 8q/ 8x fx,(z,) u
The main advantage of this setup is, that the model used for the observer design is exactly known. Since in addition to that the disturbance Le. the error-injection is known an exact linearization of the observer dynamics is possible despite any modeling errors wrt. the real plant.
(11 )
w/
and (11) is a function explicitly involving u. Now an algebraic control-law for w/ can be formulated as
The steady state accuracy of the state feed back and the observer is limited by the steady state error of the observer. This error can be rather large for some disturbances. To overcome this drawback the exactly linear system consisting of state feedback and observer can now be controlled by simple linear PI-controllers since the new linear inputs v and the controlled outputs ware decoupled. The res~lting control-structure is shown in Fig. 2.
(12) introducing the new artificial linear input VI . Since not the complete dynamics enter into eq. (12) there remains an unobservable part called zero dynamics which is required to be stable. It is sufficient to restrict the discussion to the case r = 1 since this is the case for the model under consideration. A simple extension of (12) is the case that the output is an algebraic variable Le. a component of z. In this case eq. (9) has to be differentiated implicitly wrt. t and solved for z yielding
8k u.) . f z. = (8k)-1 8z (8k ax+8u
(13)
Here the jacobian of eq. (9) wrt. z has to be inverted which is only possible if it has full rank and hence (8),(9) needs to be an index 1 differential algebraic equation system. The control law finally reads for T/ = 1 0=
(8k) -1 8kf - (w/D -Wl) 8z -8z ax
8q/
Tl-
K/Vl.
(14)
Fig. 2. Signal flow for asymptotically exact input/output-linearization
In combination with observers an important class of disturbances d enters additive into eq. (8) yielding
x = f(x,z, u) + d .
(15)
4. CONTROL LAW FOR THE COLUMN
If d is known the control law can be extended by replacing f with f + d yielding e.g. for eq. (14) 0=
T/
(8k) -1(8k (f + 8z 8z ax
8ql
- (wP -
The objective of the control for the column under consideration is to keep the temperatures T4 and T60 at their setpoints TP and T£ respectively and thereby control the top and bottom product composition. The temperature of a specific tray in the column can be obtained form the boiling point condition (5) where it appears as an implicit algebraic variable. Implicit differentiation and transformation to the form of equation (14) yields
d)) Wl) -
K/v/
(16)
and, since the disturbance is explicitly included, it is automatically rejected.
3.2 Asymptotically exact input/output-linearization In order to implement either of the control laws (12), (14) or (16) the state of the plant needs to be known. For this reason an observer is needed to reconstruct the state of the plant from the available measurements. Now it is quiet natural
lIS
for k = {4, 60} and the expression in the outmost brackets is Tk. In the remainder of this paper the time constants Tj and the gains Kl are fixed at 1 and the linear controllers are designed as PI-controllers with first order delay, compensating the pole of the linearization. The other two poles are both placed at -2.
10
10
10
20
20
20
E30
E30
E30
40
40
50
50
00
60
340
330
>00
error injection
0.6
0.8
componentS
..
00
40
50
00 ""'-:,.".,...--,.,-_ _---' 0.02
0.04
0.011
0.08
0.02 0.04 0.011 0.08
componentD
componentE
Fig. 3. Steady state profiles for temperature and composition
As mentioned before the key component of the presented control-structure is the underlying observer. The observer to be considered here is based on a simulator augmented by an error injection (Zeitz, 1977; Mangold et al., 1994). This results in
simulator
0.4
10
5. OBSERVER DESIGN
0= k(x, z, u), w= q(x,i), y = b(x, i, u).
.. .
20
componentC
--...,..-......
0.6
component A
60<---0.0"-,- 0.1-0."-'
"-v-"
0.4
0.2
360
temperature
Although eq. (17) looks pretty compact one should be aware of the fact that the complete thermodynamics are involved in the control law hidden in the partial derivatives of eq. (4) makin~ eq. (17) highly nonlinear.
,l=f(x,z,u)+ G(y-y),
40 50
Fig. 3. They show, that in the vicinity of tray k = 4 the temperature increases from top to bottom. In the same region Xl,k is decreasing considerably. From this observation it can be concluded, that if the measured temperature T4 is higher than the estimated T4 Xl,4 has to be reduced and hence G 1 ,4 = o. This procedure is now repeated for all elements Gi,k, k = {4,60}, i = l(l)NC 1 setting the unsure elements to zero, finding G 1,4 = +0, G 2,4 = -0, G 4,60 = +0 and all other
(18) (19)
Gi,k
(20)
=
o.
Due to the coupling of observer and controller dynamics the final magnitude of 0 can be determined in closed loop simulation studies only.
(21)
Further appropriate initial conditions Xo together with 0 = k(Xo, i o, uo) have to be given. Eq. (21) models the measurement.
5.1 Observer simulation studies
The observer design consists of two main tasks. The first one is the determination of the number of sensors together with their placement along the column and the second is the design of the coupling matrix G.
With the simulation studies presented in Fig. 4 the dependence of the observer dynamics upon 0 is illustrated. Of special interest are the settling time of the observer and the steady state error.
Considerations based on simulation studies and analysis of the column profile dynamics suggest that the controlled temperatures T4 and T60 are good choices. This is supported by considering the control law (17) and remarking that the concentrations Xi,k on the corresponding tray have an important role. Hence the observer should yield good approximations for at least those trays. This is facilitated by exact knowledge of the observation error y - y obtained from a direct measurement in contrast to a reconstruction of the error profile from measurements on other trays.
In Fig. 4 on the left half the settling time is investigated by varying o. The observer is started with initial conditions xo, Zo corresponding to a steady state obtained for Q perturbed by +10% while the process is at its nominal operating point. At t = the disturbance to the observer is
°
344.'
y r-_tra-.:._4-----,
tray 60
.....
tray 4
tray 60
....•t::===I==1
303.5
Next the G matrix is designed. On the same basis as the sensor placement the error injection is only implemented on the controlled trays, yielding ±i,k = fi,k(X , i, u) + Gi ,k (ih - Yk) , k = {4, 60}, i = l(l)NC - 1
313.4 __ _' _ . _ . _ . _ .
""'Oc-.,.. , -:":"--',,
(22)
time
with Gi,k E {-o, 0, +o}, 0 E IR and all other elements set to zero. The steady state profiles of temperatures and compositions are depicted in
o~-,-'0--"5 time
J.i°o'---,-'0--"5 time
0
5
10
15
time
Fig. 4. Dependance of the observer dynamics on o . The plots show 0 = {a, 1,2,4,8, 16}. 116
removed. Comparison to the case Q = 0, Le. a pure simulator, clarifies the effect of the error injection and it shows that the settling time decreases with increasing Q .
not able to completely decouple T60 from T4 while the nonlinear controller almost eliminates the coupling even in the presence of the unknown input delays. Due to the better decoupling the nonlinear controller brings the manipulated variables earlier to rest than the linear controller.
Next, the steady state error is investigated. The corresponding results are presented in Fig. 4 on the right half. Again the same initial conditions are used but the disturbance is not removed. As before the results improve with increased Q.
6.2 Disturbance rejection
The complete error profiles along the column do not show the same excellent convergence to the plant state as the temperatures on the measurement trays. Important is the fact that the concentrations entering into the control law eq. (17) exhibit the same excellent convergence properties as the temperatures in Fig. 4. This result gives full confidence that the observer is capable of giving good estimates for the controller.
In order to examine the disturbance rejection four step changes have been considered: subcooling in the condenser of lOK, feed increase of 10%, feed reduction of 10% and a change in feed composition. The transients generated by these disturbances wrt. the bottom key component are shown in Fig. 6. Evidently the nonlinear controller han...tJooaIIrGoIlOK
0 .01
I ! iO.. ; .,.•. _ .... jO_0'7
....
f
" 313.85 ....
o
o
. , .,........
.:
Q
--,
... Hi\. H I::::: L
.
,.
.
.: .
,
15
20
.. ...•
.. .....
31.1'.""- - - , - .--,,----'..
7. CONCLUSIONS
'4.2r----r-=="'~_=rR--,
,... .... .1.. =:1 . .
t ,... .•..
V
....
10
~
....r----r-====...=-=:::f
I.. , ,. .
....
j .....
to'$
J..~: ..... , ~ ~. ..
5
In Fig. 7 the controlled and manipulated variables are displayed for the case of a feed reduction of 10% (cf. Fig. 6, bottom left) and although the nonlinear controller brings the plant. faster to rest it does not need larger amplitudes on the manipulated variables than the linear controller .
.. u,...--,.--,---..:.----,
----~
.:
,
dies the disturbances by far better than the linear controller. For the top key component both controllers show good performance with advantages for the nonlinear controller, most evidently for the subcooling however the results are not shown for brevity.
~~ .
.
0.2 .......\.~~'
..
Fig. 6. Disturbance rejection wrt. bottom key component
, ".8 .. • .. • ..
113.58 ....
lo... · JI • .. ..... ; ,
.L.:.'...::,_--'-___...J
.
1 .... '1-- == ..... ...
I::·/T~~.2:-r·· ====., I .. •• ..... ..
..
oanboIIId . . . . . . T.
I,
.. .
J 0 .01 .. ~ . ....~.,~ ....
For mimo-systems such as the plant under consideration decoupling of the controlled variables is desirable. In Fig. 5 the transients of controlled
I, .
.. .
1:IK~~f ~o l .H' : .H '.
6.1 Setpoint changes
·· . t
I·
'" : :
O...,.L~"~---,o--'-_...J ..
At the real plant so far unmodeled delays on the inputs Rand Q are present and can be approximated by first order delays. In order to show the robustness of the controllers regarding such additional delays they are implemented for all following simulation studies. Furthermore the use of the simplified design model (1)-(7) is justified by using a detailed model for the column that particularly includes rigorous hydrodynamics.
38S.7
I
0 ..
In order to evaluate the performance of the nonlinear controller it is compared to a well tuned lower triangular linear mimo controller. For details on the linear controller see (Fernholz et al., 2001).
OOI"ItPoIedvaMtM T.,
·
==...:.. 11 .
6. CLOSED-LOOP PERFORMANCE
....71r-:::.-,.--,---~---,
··
I'
A nonlinear control scheme, consisting of state feed back and observer has been designed. It is based on temperature measurements and hence easily applicable to the real plant. In simulation studies the nonlinear controller exhibits excellent performance compared to a well tuned linear controller, even in the presence of unknown input delays thereby proofing its robustness. The improved performance is not obtained at the price of higher amplitudes on the inputs, as might be
....
.i " ., . 1'1...
l.I----.....
'02 --1 ,.,'--_ _ _ _ _- - l
•
.... '0
16
20
Fig. 5. Setpoint changes of ±1K on tray 4 and manipulated variables are shown for set point changes of ±1K on T4 • The linear controller is 117
Greek letters
conWIIId ...1W:I6I T. 72!r-----,.-~-~_
t .. ..
fi
Q
'/~;;;';~~;;;"~:;"~;;;;~'
. r··,,, -~ I
'Y
-noftIn. oont.
81 . , ......
-
_.,..,oarc. .........
:
.
11
:
T
·'0""--;'--''-0-
.....
.
.
Sub- and superscripts D F
.
.....
\
28 . .\
j
-\ ..
..0....---'-"' .... "--''-0-
.....
..... '.---'20
..... 10
15
k 1 L s
20
Fig. 7. Controlled and manipulated variables for a feed reduction of 10%
V
expected, but is realized by utilizing the intrinsic process dynamics incorporated into the controller by the state feed back.
Fernholz, G., M. Friedrich, S. Gruner, K.-D . Mohl, A. Kienle and E.D. Gilles (2001). Control structure selection and linear multiple-in putmultiple-output controller desig for an industrial reactive distillation column. Submitted to DYCOPS 6. Friedrich, M., M. Storz and E.D. Gilles (1992). In: Preprints Dycord+ '92 - IFAC Symposium. IFAC. University of Maryland, College Park, USA. pp. 369-374. Groebel, M., F. Allgower, M. Storz and E. D. Gilles (1995). In: Proc. ACC' 95. pp. 26482652. Groebel, M.J. (1996). Asymptotisch exakte Ein/ Ausgangs-Linearisierung von Destillationskolonnen. PhD thesis. Institiut fur Systemdynamik und Regelungstechnik, Universitat Stuttgart. Isidori, A. (1995). Nonlinear Control Systems. 3rd edition ed .. Springer-Verlag. Kumar, A. and P. Daoutidis (1999). AIChE Jounal45(1), 51- 68. Mangold, M., G. Lauschke, J . Schaffner, M. Zeitz and E.D. Gilles (1994). J. Proc. Cont. 4(3), 163-172. Monroy-Loperena, R., E. Perez-Cisneros and J. Alvarez-Ramfrez (1999). Nonlinear PI Control of an Ethylene Glycol Reactive Distillation Column. Comput. Chem. Engng. Suppl. pp. 835-838. Nijmeijer, H. and A.J. van der Schaft (1990). Nonlinear Dynamical Control Systems. SpringerVerlag. Ried, R. C., J. M. Prausnitz and B. E. Poling (1987) . The Properties of Gases and Liquids. McGraw-Hill. New York. Sneesby, M.G., M.O. Tade and T .N. Smith (1999). J. Proc. Contr. 9, 19-31. Zeitz, M. (1977) . Nichtlineare Beobachter fur chemische Reaktoren. VDI-Verlag GmbH Dusseldorf.
The authors express their thanks to the 'Bundesministerium fur Bildung, Wissenschaft, Forschung und Technologie' for sponsoring this work within the research project 'Analyse und Synthese mehrphasiger Reaktionssyteme mit Methoden der nichtlinearen Dynamik - Teil B' (Sponsoring ID 03 C 0268 B).
9. NOTATION Latin letters
f
h
k K L
n NC NR p q r ro
u V v Vol w x y
z
dynamic equation observer gain molar enthalpy section 5: measurement eq. algebraic equation gain in linearization liquid flow rate molar holdup number of components number of reactions pressure output equation relative (vector) degree reaction rate manipulated variable vapour flow rate linear input volume holdup controlled variable fluid phase composition section 3: dynamic state vapour phase composition section 5: measurement algebraic state
indicates setpoints (desired) related to the feed component index reaction index tray index output index liquid saturation vapour
10. REFERENCES
8. ACKNOWLEDGMENTS
G
[l/(sK)]
[-] [-] [s]
..... '.---'20
)., \ . .:j::.:. :":..j'H HHH 130 . ., ,, :. --lineuoonl.
observer gain activity coefficient stoichiometric coefficient time constant
[J/mol]
[-] [mol/s] [mol]
[-] [-] [bar]
[-] [mol/s] [m3 ]
[-]
[-]
118
Copyright !Cl IF AC Dynamics and Control of Process Systems, Jejudo Island, Korea, 200 I
C:UC> Publications www.elsevier.comllocate/ifac
NONLINEAR CONTROL OF AN INDUSTRIAL REACTIVE DISTILLATION COLUMN S. Griiner * K.-D. Mohl * A. Kienle ** E. D. Gilles ., •• G. Fernholz"· M. Friedrich *** • Universitat Stuttgart, Institut fUr Systemdynamik und Regelungstchnik, Pfaffenwaldring g, 70550 Stuttgart, Germany . .. Max-Planck-Institut fUr Dynamik Komplexer Technischer Systeme Magdebury, Zenit-Gebaude, Leipziger Strafle 44, 39120 Magdebury, Germany **. Bayer A G, Process Technology, Process Control, 51368 Leverkusen, Germany
Abstract: Control of reactive distillation columns is a challenging task due to the complex dynamics arising from the coupling of reaction and separation. In this paper asymptotically exact input/output-linearization is applied to an industrial reactive distillation column. This control scheme requires knowledge of the complete state of the process and therefore an observer is designed. To compensate for steady state observer offsets an outer control loop with simple PI-controllers is implemented. In comparison to a well tuned linear controller the nonlinear controller shows a superior performance with respect to setpoint-changes and disturbances, even in the presence of unknown input delays. Copyright © 2001lFAC Keywords: nonlinear control, observers, robustness, reactive distillation columns
active distillation column, including the observer design, based on simple temperature measurements is considered. After a brief description of the process model the basic idea of the control concept, namely input/output-linearization is presented and applied to the process. Then a suitable observer is designed and tested in simulation studies. Finally the robust performance of the nonlinear controller is compared to that of a well tuned linear controller (Fernholz et al., 2001) wrt. set point changes and disturbance rejection.
1. INTRODUCTION
Reactive distillation has recieved a tremendous industrial interest over the last years. Many publications have been made regarding the design of such columns, however concentrating on the steady state behaviour. Even more striking are the complex dynamics of reactive distillation columns, due to the coupling of reaction and separation. Nevertheless only few publications are available regarding the control of reactive distillation columns and most of them consider only linear control e.g. (Sneesby et al., 1999). Those employing nonlinear controllers assume that the state of the plant is completly measurable and thereby neglect the need to design an observer that performs well in the closed loop e.g. (Kumar and Daoutidis, 1999). Others rely on concentration measurements that are rather difficult to obtain on a real plant e.g. (Monroy-Loperena et al., 1999) .
The process under consideration is an industrial reactive distillation column schematically shown in Fig. 1. The column consists of NS = 63 trays including the reboiler as tray NS and the condenser as tray 1.
In this contribution, to our knowledge for the first time, nonlinear control of an industrial re-
The feed enters on tray kF = 25 and contains the components Band C from which the product E
2. PROCESS
113
Ne
0= 1- LYi,k
top key component: C
(5)
;=1
B.CL F ~ . ", liquid reed f-~-
A+E"'B+D
.
:
holds. The pressure is computed according to 0= Pk - D..p - Pk-l , PI = const. (6) and the model is completed by the summation condition for the liquid-phase mole fractions
Cbemical Reactioas:
A+D"'B+C
~-T.\
Ne 0= 1- LXi,k. i=1
k k+1
The main nonlinearities are introduced by the thermodynamic quantities. Namely the activity coefficients li,k, computed according to Wilson and the saturation pressures Ps,i, computed according to Antoine and a polynomial approach depending on the component (Ried et al., 1987). Another non linearity are the reaction rates rO,j,k which are based on an Arrhenius-like formulation.
bonom product E bonom key component: 0
Q
Fig. 1. Schematic view of the column is produced. In the fluid phase NR = 2 reactions with reaction rates rO,j,k occur and the simplified reaction scheme is given in Fig. 1. The lower boiling component A is removed as the top product and its purity is specified in terms of a threshold for the top key component, the impurity C. The main product E is removed form the column as bottom product and its purity is specified in terms of thresholds for the bottom key component, the impurity D.
For the reboiler (k = NS) the balances (1)-(3) are adjusted since there is no VNS+1 and the energy balance (3) is extended by the additive term Q representing the reboiler heat input. The material balances (1) and (2) are replaced for the condenser (k = 1) by equations stating no composition change from tray k = 2 and equal liquid and fluid composition. The energy balance (3) is dropped and replaced by the equation for the reflux ratio R = Lt/VI'
Previous results reported in (Groebel et al., 1995) for non reactive distillation indicate that there is no need to explicitly include fluid dynamics in t.he design model. Hence the model is based on the assumption of constant volume holdup and therefore the molar holdup nk depends on the composition. For each of the ordinary trays k = 2(1)NS -1 the model consists of the dynamic component material balances dx· k nk d~'
= Lk-lX;,k-l -
The plant model consists of NS· (NC -1) dynamic and NS· (NC + 5) algebraic equations corresponding to 252 dynamic and 630 algebraic states. 3. BASIC IDEA OF INPUT /OUTPUT-LINEARIZATION
LkX;,k
NR
Instead of a detailed presentation of the underlying theory only a brief sketch of the concept is given and emphasis is on the application. For details on the theory the reader is refered to (Isidori, 1995; Nijmeijer and van der Schaft, 1990) or (Friedrich et al., 1992) . An application to nonreactive distillation can be found in (Groebel, 1996; Groebel et al., 1995).
xn '
F +Vk+1Yi,k+l- VkYi,k+ LVolkvi,jrO,j,k+(L j=l i = l(l)NC - 1.
(1)
The flow rates are computed from the stationary total material balance
The system under consideration can be represented by
0= Lk-l - Lk + Vk+1 - Vk + (LF) Ne NR L Volkvi,jrO,j,k
+L
(2)
x = f(x, z, u),
;=1 j=l
0= k(x, z, u),
and the stationary energy balance
+ Vk+lhV,k+l VkhV,k + (LFh F) .
w
0= Lk-lhL,k-l - LkhL,k -
= q(x,z)
(8) (9) (10)
for t > 0 and the initial conditions x(t = 0) = Xo together with 0 = k(Xo,zo,uo), x E JRn,z E JRI and u, W E JRm i.e. a quadratic multi-input/ multi-output (mimo) system.
(3)
Since standard enthalpies of formation are used, no reaction term appears in (3) . In eq. (1)-(3) the terms in brackets appear only for the feed tray. Besides the vapour-liquid equilibrium relation Ps,i O = Y; ,k - - , i,kXi,k Pk
(7)
3.1 Derivation of control law
(4)
The idea of input/output-linearization is to differentiate the elements Wl wrt. time t, replacing all
the boiling point condition
114
terms x by f(x, z, u) until an expression explicitly involving at least one component of u is obtained. The number of differentiations needed is called relative degree T and for mimo systems the vector r = h, T2 , ... , TmJ is called relative vector degree.
to linearize the observer dynamics instead of the plant dynamics and to interpret the error injection of the observer as a disturbance. Due to the asymptotic convergence of the observer to the plant an asymptotically exact input/outputlinearization is obtained.
For systems with T/ = 1 and q/ depending only on the differential variables x this procedure yields . = 8q/ 8x fx,(z,) u
The main advantage of this setup is, that the model used for the observer design is exactly known. Since in addition to that the disturbance Le. the error-injection is known an exact linearization of the observer dynamics is possible despite any modeling errors wrt. the real plant.
(11 )
w/
and (11) is a function explicitly involving u. Now an algebraic control-law for w/ can be formulated as
The steady state accuracy of the state feed back and the observer is limited by the steady state error of the observer. This error can be rather large for some disturbances. To overcome this drawback the exactly linear system consisting of state feedback and observer can now be controlled by simple linear PI-controllers since the new linear inputs v and the controlled outputs ware decoupled. The res~lting control-structure is shown in Fig. 2.
(12) introducing the new artificial linear input VI . Since not the complete dynamics enter into eq. (12) there remains an unobservable part called zero dynamics which is required to be stable. It is sufficient to restrict the discussion to the case r = 1 since this is the case for the model under consideration. A simple extension of (12) is the case that the output is an algebraic variable Le. a component of z. In this case eq. (9) has to be differentiated implicitly wrt. t and solved for z yielding
8k u.) . f z. = (8k)-1 8z (8k ax+8u
(13)
Here the jacobian of eq. (9) wrt. z has to be inverted which is only possible if it has full rank and hence (8),(9) needs to be an index 1 differential algebraic equation system. The control law finally reads for T/ = 1 0=
(8k) -1 8kf - (w/D -Wl) 8z -8z ax
8q/
Tl-
K/Vl.
(14)
Fig. 2. Signal flow for asymptotically exact input/output-linearization
In combination with observers an important class of disturbances d enters additive into eq. (8) yielding
x = f(x,z, u) + d .
(15)
4. CONTROL LAW FOR THE COLUMN
If d is known the control law can be extended by replacing f with f + d yielding e.g. for eq. (14) 0=
T/
(8k) -1(8k (f + 8z 8z ax
8ql
- (wP -
The objective of the control for the column under consideration is to keep the temperatures T4 and T60 at their setpoints TP and T£ respectively and thereby control the top and bottom product composition. The temperature of a specific tray in the column can be obtained form the boiling point condition (5) where it appears as an implicit algebraic variable. Implicit differentiation and transformation to the form of equation (14) yields
d)) Wl) -
K/v/
(16)
and, since the disturbance is explicitly included, it is automatically rejected.
3.2 Asymptotically exact input/output-linearization In order to implement either of the control laws (12), (14) or (16) the state of the plant needs to be known. For this reason an observer is needed to reconstruct the state of the plant from the available measurements. Now it is quiet natural
lIS
for k = {4, 60} and the expression in the outmost brackets is Tk. In the remainder of this paper the time constants Tj and the gains Kl are fixed at 1 and the linear controllers are designed as PI-controllers with first order delay, compensating the pole of the linearization. The other two poles are both placed at -2.
10
10
10
20
20
20
E30
E30
E30
40
40
50
50
00
60
340
330
>00
error injection
0.6
0.8
componentS
..
00
40
50
00 ""'-:,.".,...--,.,-_ _---' 0.02
0.04
0.011
0.08
0.02 0.04 0.011 0.08
componentD
componentE
Fig. 3. Steady state profiles for temperature and composition
As mentioned before the key component of the presented control-structure is the underlying observer. The observer to be considered here is based on a simulator augmented by an error injection (Zeitz, 1977; Mangold et al., 1994). This results in
simulator
0.4
10
5. OBSERVER DESIGN
0= k(x, z, u), w= q(x,i), y = b(x, i, u).
.. .
20
componentC
--...,..-......
0.6
component A
60<---0.0"-,- 0.1-0."-'
"-v-"
0.4
0.2
360
temperature
Although eq. (17) looks pretty compact one should be aware of the fact that the complete thermodynamics are involved in the control law hidden in the partial derivatives of eq. (4) makin~ eq. (17) highly nonlinear.
,l=f(x,z,u)+ G(y-y),
40 50
Fig. 3. They show, that in the vicinity of tray k = 4 the temperature increases from top to bottom. In the same region Xl,k is decreasing considerably. From this observation it can be concluded, that if the measured temperature T4 is higher than the estimated T4 Xl,4 has to be reduced and hence G 1 ,4 = o. This procedure is now repeated for all elements Gi,k, k = {4,60}, i = l(l)NC 1 setting the unsure elements to zero, finding G 1,4 = +0, G 2,4 = -0, G 4,60 = +0 and all other
(18) (19)
Gi,k
(20)
=
o.
Due to the coupling of observer and controller dynamics the final magnitude of 0 can be determined in closed loop simulation studies only.
(21)
Further appropriate initial conditions Xo together with 0 = k(Xo, i o, uo) have to be given. Eq. (21) models the measurement.
5.1 Observer simulation studies
The observer design consists of two main tasks. The first one is the determination of the number of sensors together with their placement along the column and the second is the design of the coupling matrix G.
With the simulation studies presented in Fig. 4 the dependence of the observer dynamics upon 0 is illustrated. Of special interest are the settling time of the observer and the steady state error.
Considerations based on simulation studies and analysis of the column profile dynamics suggest that the controlled temperatures T4 and T60 are good choices. This is supported by considering the control law (17) and remarking that the concentrations Xi,k on the corresponding tray have an important role. Hence the observer should yield good approximations for at least those trays. This is facilitated by exact knowledge of the observation error y - y obtained from a direct measurement in contrast to a reconstruction of the error profile from measurements on other trays.
In Fig. 4 on the left half the settling time is investigated by varying o. The observer is started with initial conditions xo, Zo corresponding to a steady state obtained for Q perturbed by +10% while the process is at its nominal operating point. At t = the disturbance to the observer is
°
344.'
y r-_tra-.:._4-----,
tray 60
.....
tray 4
tray 60
....•t::===I==1
303.5
Next the G matrix is designed. On the same basis as the sensor placement the error injection is only implemented on the controlled trays, yielding ±i,k = fi,k(X , i, u) + Gi ,k (ih - Yk) , k = {4, 60}, i = l(l)NC - 1
313.4 __ _' _ . _ . _ . _ .
""'Oc-.,.. , -:":"--',,
(22)
time
with Gi,k E {-o, 0, +o}, 0 E IR and all other elements set to zero. The steady state profiles of temperatures and compositions are depicted in
o~-,-'0--"5 time
J.i°o'---,-'0--"5 time
0
5
10
15
time
Fig. 4. Dependance of the observer dynamics on o . The plots show 0 = {a, 1,2,4,8, 16}. 116
removed. Comparison to the case Q = 0, Le. a pure simulator, clarifies the effect of the error injection and it shows that the settling time decreases with increasing Q .
not able to completely decouple T60 from T4 while the nonlinear controller almost eliminates the coupling even in the presence of the unknown input delays. Due to the better decoupling the nonlinear controller brings the manipulated variables earlier to rest than the linear controller.
Next, the steady state error is investigated. The corresponding results are presented in Fig. 4 on the right half. Again the same initial conditions are used but the disturbance is not removed. As before the results improve with increased Q.
6.2 Disturbance rejection
The complete error profiles along the column do not show the same excellent convergence to the plant state as the temperatures on the measurement trays. Important is the fact that the concentrations entering into the control law eq. (17) exhibit the same excellent convergence properties as the temperatures in Fig. 4. This result gives full confidence that the observer is capable of giving good estimates for the controller.
In order to examine the disturbance rejection four step changes have been considered: subcooling in the condenser of lOK, feed increase of 10%, feed reduction of 10% and a change in feed composition. The transients generated by these disturbances wrt. the bottom key component are shown in Fig. 6. Evidently the nonlinear controller han...tJooaIIrGoIlOK
0 .01
I ! iO.. ; .,.•. _ .... jO_0'7
....
f
" 313.85 ....
o
o
. , .,........
.:
Q
--,
... Hi\. H I::::: L
.
,.
.
.: .
,
15
20
.. ...•
.. .....
31.1'.""- - - , - .--,,----'..
7. CONCLUSIONS
'4.2r----r-=="'~_=rR--,
,... .... .1.. =:1 . .
t ,... .•..
V
....
10
~
....r----r-====...=-=:::f
I.. , ,. .
....
j .....
to'$
J..~: ..... , ~ ~. ..
5
In Fig. 7 the controlled and manipulated variables are displayed for the case of a feed reduction of 10% (cf. Fig. 6, bottom left) and although the nonlinear controller brings the plant. faster to rest it does not need larger amplitudes on the manipulated variables than the linear controller .
.. u,...--,.--,---..:.----,
----~
.:
,
dies the disturbances by far better than the linear controller. For the top key component both controllers show good performance with advantages for the nonlinear controller, most evidently for the subcooling however the results are not shown for brevity.
~~ .
.
0.2 .......\.~~'
..
Fig. 6. Disturbance rejection wrt. bottom key component
, ".8 .. • .. • ..
113.58 ....
lo... · JI • .. ..... ; ,
.L.:.'...::,_--'-___...J
.
1 .... '1-- == ..... ...
I::·/T~~.2:-r·· ====., I .. •• ..... ..
..
oanboIIId . . . . . . T.
I,
.. .
J 0 .01 .. ~ . ....~.,~ ....
For mimo-systems such as the plant under consideration decoupling of the controlled variables is desirable. In Fig. 5 the transients of controlled
I, .
.. .
1:IK~~f ~o l .H' : .H '.
6.1 Setpoint changes
·· . t
I·
'" : :
O...,.L~"~---,o--'-_...J ..
At the real plant so far unmodeled delays on the inputs Rand Q are present and can be approximated by first order delays. In order to show the robustness of the controllers regarding such additional delays they are implemented for all following simulation studies. Furthermore the use of the simplified design model (1)-(7) is justified by using a detailed model for the column that particularly includes rigorous hydrodynamics.
38S.7
I
0 ..
In order to evaluate the performance of the nonlinear controller it is compared to a well tuned lower triangular linear mimo controller. For details on the linear controller see (Fernholz et al., 2001).
OOI"ItPoIedvaMtM T.,
·
==...:.. 11 .
6. CLOSED-LOOP PERFORMANCE
....71r-:::.-,.--,---~---,
··
I'
A nonlinear control scheme, consisting of state feed back and observer has been designed. It is based on temperature measurements and hence easily applicable to the real plant. In simulation studies the nonlinear controller exhibits excellent performance compared to a well tuned linear controller, even in the presence of unknown input delays thereby proofing its robustness. The improved performance is not obtained at the price of higher amplitudes on the inputs, as might be
....
.i " ., . 1'1...
l.I----.....
'02 --1 ,.,'--_ _ _ _ _- - l
•
.... '0
16
20
Fig. 5. Setpoint changes of ±1K on tray 4 and manipulated variables are shown for set point changes of ±1K on T4 • The linear controller is 117
Greek letters
conWIIId ...1W:I6I T. 72!r-----,.-~-~_
t .. ..
fi
Q
'/~;;;';~~;;;"~:;"~;;;;~'
. r··,,, -~ I
'Y
-noftIn. oont.
81 . , ......
-
_.,..,oarc. .........
:
.
11
:
T
·'0""--;'--''-0-
.....
.
.
Sub- and superscripts D F
.
.....
\
28 . .\
j
-\ ..
..0....---'-"' .... "--''-0-
.....
..... '.---'20
..... 10
15
k 1 L s
20
Fig. 7. Controlled and manipulated variables for a feed reduction of 10%
V
expected, but is realized by utilizing the intrinsic process dynamics incorporated into the controller by the state feed back.
Fernholz, G., M. Friedrich, S. Gruner, K.-D . Mohl, A. Kienle and E.D. Gilles (2001). Control structure selection and linear multiple-in putmultiple-output controller desig for an industrial reactive distillation column. Submitted to DYCOPS 6. Friedrich, M., M. Storz and E.D. Gilles (1992). In: Preprints Dycord+ '92 - IFAC Symposium. IFAC. University of Maryland, College Park, USA. pp. 369-374. Groebel, M., F. Allgower, M. Storz and E. D. Gilles (1995). In: Proc. ACC' 95. pp. 26482652. Groebel, M.J. (1996). Asymptotisch exakte Ein/ Ausgangs-Linearisierung von Destillationskolonnen. PhD thesis. Institiut fur Systemdynamik und Regelungstechnik, Universitat Stuttgart. Isidori, A. (1995). Nonlinear Control Systems. 3rd edition ed .. Springer-Verlag. Kumar, A. and P. Daoutidis (1999). AIChE Jounal45(1), 51- 68. Mangold, M., G. Lauschke, J . Schaffner, M. Zeitz and E.D. Gilles (1994). J. Proc. Cont. 4(3), 163-172. Monroy-Loperena, R., E. Perez-Cisneros and J. Alvarez-Ramfrez (1999). Nonlinear PI Control of an Ethylene Glycol Reactive Distillation Column. Comput. Chem. Engng. Suppl. pp. 835-838. Nijmeijer, H. and A.J. van der Schaft (1990). Nonlinear Dynamical Control Systems. SpringerVerlag. Ried, R. C., J. M. Prausnitz and B. E. Poling (1987) . The Properties of Gases and Liquids. McGraw-Hill. New York. Sneesby, M.G., M.O. Tade and T .N. Smith (1999). J. Proc. Contr. 9, 19-31. Zeitz, M. (1977) . Nichtlineare Beobachter fur chemische Reaktoren. VDI-Verlag GmbH Dusseldorf.
The authors express their thanks to the 'Bundesministerium fur Bildung, Wissenschaft, Forschung und Technologie' for sponsoring this work within the research project 'Analyse und Synthese mehrphasiger Reaktionssyteme mit Methoden der nichtlinearen Dynamik - Teil B' (Sponsoring ID 03 C 0268 B).
9. NOTATION Latin letters
f
h
k K L
n NC NR p q r ro
u V v Vol w x y
z
dynamic equation observer gain molar enthalpy section 5: measurement eq. algebraic equation gain in linearization liquid flow rate molar holdup number of components number of reactions pressure output equation relative (vector) degree reaction rate manipulated variable vapour flow rate linear input volume holdup controlled variable fluid phase composition section 3: dynamic state vapour phase composition section 5: measurement algebraic state
indicates setpoints (desired) related to the feed component index reaction index tray index output index liquid saturation vapour
10. REFERENCES
8. ACKNOWLEDGMENTS
G
[l/(sK)]
[-] [-] [s]
..... '.---'20
)., \ . .:j::.:. :":..j'H HHH 130 . ., ,, :. --lineuoonl.
observer gain activity coefficient stoichiometric coefficient time constant
[J/mol]
[-] [mol/s] [mol]
[-] [-] [bar]
[-] [mol/s] [m3 ]
[-]
[-]
118