Design, analysis and real zation of a state control system for an industrial ( istillation column G. Fieg, G. Wozny
and L. Jeromin
Henkel KGaA, 4000 Diisseldorf, Postfach 1100, Germany (Received 14 February 1991; revised 24 July 1992)
A systematic method for the development and practical implementation of a control system on the basis of a state observer and controller for the bottom concentration of an industrial distillation column is described. The development and testing of the state observer and state controller together with the system identification were carried out exclusively on the basis of simulation calculations. Initially the control system was studied in short individual tests, before a final long-term test was carried out under operating conditions of the real unit. The experience obtained in implementing the state control system is described. It was found that, with regard to dynamic column behaviour during disturbances, the state control system was superior to conventional PI-control concepts. (Keywords: distillation; state observer; state controller; methanol)
Greater competition and stricter regulations with regard to plant safety and environmental protection mean that an increasing number of chemical plants are operated under very restricted conditions. To make matters worse, process variables that are of importance for process control and plant safety are often not directly measurable, or can only be measured after laborious sample preparation and laboratory analyses involving considerable dead time. Typical examples of such variables are the concentrations of chemical products and characteristic features of polymers, such as chain length distribution etc. In this context, optimal plant control from an econl omit point of view can only be successfully achieved by obtaining detailed information of the process variables and by applying advanced control methods. It is essential to develop mathematical models that contain as much a priori knowledge of the process as possible, so that model-based predictions retain their validity over wide ranges of parameter variation. This is especially important for processes involving intensive material and energy coupling. A systematic method of developing and putting into operation such a model-based control concept, which relies on the use of a state observer and controller, is described below. It was implemented in an industrial distillation column with a sidestream used by Henkel KGaA, Dusseldorf, Germany, for purification of methanol-containing waste-water. The advantage of this method is that the whole development can be carried out exclusively on the basis of
*Author
stationary and dynamic simulation. The real plant is not necessary at this stage. In addition, this method allows the control system to be tested repeatedly, and thus made more secure, before it is attached on-line to the distillation column. Silberberger’, Gilles and Retzbach’ and Retzbach3 have applied state-space control to different extractive distillation plants such as single columns with or without sidestream and to a two-column system. Gilles et ~1.~and Eckelmann’ tested these concepts with an industrial
to whom all correspondence should be addressed
0959%1524/92/04017%09 0 1992 Butterworth-Heinemann
Ltd
179
State control system for an industrial distillation column: G. Fieg et al.
< 2000 ppm H,O
F, *F
mo Figure 1
Methanol-water
B, *B
>
>I
L
separation
-I ~1000 ppm CH,OI column
(50 real stages)
a state controller to control the vapour temperature of industrial boilers. In comparison with conventional control concepts (with PID controllers), the state control system exhibited greatly improved control performance and was easy to implement. In the field of chemical reactors, Zeitz” has proposed design methods for non-linear state observers. With Luenberger observers and Kalman filters, valuable instruments are available to control technology for monitoring reactors and diagnosing errors”X’2. These developments link up with research work in the field of model construction, control, adjustment and, finally, monitoring of batch and semibatch polymerization reactors. Reference should be made to the overview by MacGregor et aLI3 with regard to this subject.
Description of the plant Figure 1 is a schematic representation of the distillation column, which is used to purify methanol-containing waste-water. The methanol content varies between 8 and 35% by weight. Besides methanol, other components
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such as ethanol, i- and n-propanol (l-2% by weight in each case) were detected. The distillation column consists of 50 valve trays, has a diameter of 0.8 m and is operated at atmospheric pressure. The feed input is at the 35th tray. The feed rate varies between 2.5 and 6 m3 h-’ and is dependent on other production units of the process. The column bottom does not contain a reboiler; instead, steam at 5 bar is fed directly into the column bottom. The presence of additional components leads to the formation of azeotropic mixtures in the column. A sidestream was installed at the 20th tray to extract them. Energy is recovered by means of an additional heat exchanger. The bottom product is used to heat the feed stream. Methanol is removed at the column head and is subsequently subjected to further processing. The purity requirement here is < 2000 ppm water. Practically pure water with a maximum methanol content of 1000 ppm is produced at the bottom of the column and is fed into a sewage treatment plant. Wozny et a1.l4 have already developed and tested a control concept on the basis of extensive simulation calculations and long term optimization under operation conditions. In principle this concept involved a combination of several PID controllers coupled together through the distillation column. Level control in the distillate accumulator and column bottom was achieved through the distillate and bottom product streams respectively. The heating steam rate was used to control the bottom product concentration. For this purpose, the temperature was measured at three trays (trays 39, 41 and 44) in the stripping section of the column and the mean temperature was used as an auxiliary controlled variable. The setpoint for temperature depends on the pressure profile in the column and the feed concentration; it was represented by correlation equations in the process control system. Control of the distillate concentration was achieved through the reflux rate. The setpoint for the reflux rate is given by an equation that is dependent on the feed rate, and a correction factor that takes account of the difference between the measured temperature (trays 9, 12 and 16) and the setpoint function of the temperature. This setpoint function depends on the pressure profile. The sidestream S is set on a fixed relationship to the feed stream (7.5% of the feed rate). This control concept has been successfully tested. The plant runs in a stable manner and the integral values of both product purities are maintained. Continuous tests indicated, however, that the concentration of methanol on the bottom product is kept at 60-200 ppm CH30H on safety grounds, and is thus 5-16 times lower than the permissible limit. The dynamic behaviour of the plant also poses a problem. It was observed that, under certain circumstances, the dynamic behaviour of the plant was not satisfactory. This behaviour, which expressed itself in long term fluctuations (46 h) in the heating steam rate (manipulated variable) (Figure Z), is associated with the thermal coupling between the feed stream and the bottom product stream. In this case the feed stream was
Stare conrrol sysrem for an industrial 1400
,
column:
G. Fieg et al.
the column boundaries, the purity of the products remains guaranteed. The dynamic relationship between the temperature front on the one hand, and the vapour flowing upwards through the column and the relevant disturbance variable on the other, forms the basis of the mathematical model used for the stripping section of the column. It is defined by Equations (l)-(3):
kg/h 1200
I
distillation
1000
heating steam rate 800
T, ’ ti j.=
600
T=
+ v* = K, . m,
K2(F.xF+
R-
v*)
P+s
(1) (2) (3)
400
200
0 0
2
4
6
6
10
12 h 14
time Figure 2
Heating
steam rate wws
time (PI controller)
constant (3 m3 h-l) and the feed concentration varied between 21 and 24% methanol. As an alternative control concept, a control system consisting of a state observer and a state controller was initially developed for the bottom product concentration. In the case of this control concept, attention was directed to the dynamic behaviour of the column in response to frequent disturbances in feed rate and feed concentration.
Developing and testing the state observer All concentration control systems must take account of the considerable difficulties associated with making online measurements. In most cases, measurements of this type involve a great deal of dead time and are thus not suitable for control purposes. One possible way of dispensing with these direct measurements of concentration is to use model-based methods that can calculate the concentration of the products directly from accessible plant measurements, or can calculate suitable auxiliary variables. Another problem with all control systems is posed by considerations associated with the control structure, or, to be more precise, the specification of the connection between manipulated variable and controlled variable. Relevant information can be found in the work of Skogestad and Morari”. For the column in question a mathematical model representing a compromise between the conflicting interests mentioned above is used. The model is based on a phenomenological description of the migration of mass transfer zones as originally proposed by Silberberger’ and refined by Retzbach3. These mass transfer zones are characterized by steep temperature gradients and migrate within the column as a result of disturbances. As long as the temperature fronts are far enough from
Equation (1) states that the relationship between the manipulated variable m, (heating steam rate) and the first state variable v* (vapour flow in the column) corresponds to a PT, element. Equation (2) connects the migration velocity S of the temperature front with the disturbances feed rate F, liquid mole fraction of the feed xF, and reflux rate R. This equation is derived from a methanol balance in the stripping section of the column. The molar vapour stream v* in the upper part of the column’s stripping section is nearly constant and consists only of methanol. The molar liquid stream from the distillation section is nearly the same as the recycle stream returned to the column (equimolar overflow). In the bottom of the column the methanol concentration is sufficiently small, so that the bottom stream need not appear in the methanol balance. Equation (3) gives the relationship between the state variable (here s), which cannot be measured and the temperature T measured in the column. The factor P is between - 1.3 and - 2°C stage-‘. The following problem is encountered in solving the system of equations. Initially, the parameters K,, K,, T, and P are unknown. This problem was solved by carrying out extensive dynamic and steady-state calculations with a simulation program based on a previous study16. The calculations showed that the distillation column exhibited strongly non-linear behaviour in respect to feed concentrations encountered during operation. However, the given model (Equations (l)-(3)) is valid within a specific range surrounding a stationary operating point, for which the parameters K,, K2, T, and P were determined. For this case a switch to another linear model, with its design point in the vicinity of the new stationary operating point, was suggested as a solution. In all, three observer models were developed for the total concentration range. The selection of the appropriate model depends on the feed concentration which is measured online. For normal operation with smooth movement of the feed concentration from one model’s range to another, the automatic switching from one model to another, yields no problem. The stationary operation point is simply shifted using incremental steps to better fit the model. Some difficulties arise when the feed concentration constantly fluctuates between two models. For these cases a damping algorithm was employed to reduce the constant switching between the models.
J. Proc. Conr. 1992, Vol2, No 4
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State control system for an industrial distillation column: G. Fieg et al.
achieve favourable observer.
dynamic
behaviour
of the state
Designing and testing the state controller with the help of simulation
0 x0 4
Is , ---1
(FxF
z
I-‘-
-----y
/Y1
The task of the state controller is to keep the plant at its stationary operating point despite the occurrence of disturbances. In the formulation of an optimal control system a linear relationship exists between the state variable and the manipulated variable (Equation (8)). The controller feedback matrix k gives Equations (9) and (10) u(t) = -k . x(t)
k,
=
_
(8)
T, . (s, + %I>+ 1 4
k
=
2
_
TI *SI . SII K,
* K2
(10)
The simulation model described belowI was used to test the state controller.
6 XB 4
Figure 3
Simulation model
Block diagram of the state observer
A further problem encountered in solving the system of equations was that the state of the plant is unknown when the model is switched on, so that no start values are available for solving the differential equations. Furthermore, as a result of inaccuracies in the model, the plant state and the model state can diverge. To overcome these difficulties, a state observer as described by Luenberger” was developed. ~=A.~+B~~+D~~+~o,-~)
(4)
y = CT-x
(5)
Here the deviation between the actual temperature y measured in the column and the 9 estimated by the state observer is fed back (Figure 3), so that the observer error is diminished. The observer dynamics are determined through the feedback loops g, and g2 with specified eigenvalues (Equations (6) and (7)).
-
g, = -f
(s, +
s2
+
g
.
g2)
V-9
&)
The state observer was tested by carrying out dynamic simulations with the above mentioned program. These calculations have been described and analysed by Fieg et a1.18. Corresponding stability studies were used to
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J. Proc. Cont. 1992, Vol2, No 4
The simulation model is based on conservation equations (mass balance, energy balance and equilibrium relationships). These equations were formulated for each tray as well as for the peripheral systems (condenser and reboiler). The trays were numbered from the column head to the column bottom. The equations are based on the following assumptions: ideally mixed liquid phase; thermodynamic equilibrium between the gas and liquid phase; constant pressure profile. In addition, the capacity effect of the gas phase is ignored. The influence of variable hold-up has been investigated more closely by Wozny16. The equations for the steady state were solved with the Newton-Raphson method, based on the algorithm of Naphtali and Sandholm” and extended by Ferraris2’. The stationary solution is to be understood as a starting value for the dynamic calculations.
Testing the state controller by means of simulation For purposes of testing the state control system, the model serves as a substitute for the actual plant. The state controller was linked to the simulation program through an interface (Figure 4). The control system functions as follows. A step disturbance of the feed rate or the feed concentration is simulated. The simulation program carries out dynamic calculations to determine the corresponding temporary changes in all the relevant variables. These calculated values (‘measurements’) are
Stare control system for an industrial
simulation Basis:
state
Numerical
Fx,RTm,
control
system
‘---
equation!
Method Procedure
state -
Figure 4
Communication:
simulation
program
controller
Implementing the state control system into operation in the real plant
-.-
- state control
system
al 1400
_
kg/h 1200 I manipulated variable (heating steam rate) 1000
.
--k__
0
bl
@V,
1000
__
2000 time
3000 -
2000
3000
s
5;oo
0.4 weight %
I 0,2 fraction (bottom
G. Fieg et al.
column operation while maintaining the desired concentration of the bottom product within the desired limits. In addition, it was possible to examine the influence of the control system eigenvalues on the dynamics of the control system. It must be remarked that it is almost impossible to carry out this type of study on a real plant.
solution:
Newton-Raphson Implicitly Euler
column:
system fulfils its task and ensures unhindered
program
Mass balance Energy balance Phase equilibrium
distillation
of CH$H oroduct ) 0.0
-l 0
1000
time
Figure 5 On-line test with simulation change + 15%, eigenvalues of control
S
5000
-
program. Disturbance: feed rate system (-0.015, -0.015)
transmitted to the state observer after certain time intervals. The observer equations (4) and (5) are solved numerically (Heun’s predictor-corrector method). The estimated state variables 5 are made available to the state controller. Using Equation (8), and taking account of Equations (9) and (lo), the state controller converts these state variables into the manipulated variable (heating steam rate). This variable is fed back into the simulation program and is further processed as an input variable. Figure 5 shows the simulation results for a 15% step in the feed-rate. The eigenvalues of the control system (Equations (9) and (10)) are s1 = sI1 = - 0.015 s-‘. It can be seen from Figure 5a that the manipulated variable (heating steam rate) increased very rapidly, so that the concentration of the bottom product (Figure 5b) was practically fulfilled. Analogous tests were carried out for disturbances in the feed concentration18. On the basis of these simulations it can be stated that the state control
Before the developed state control system was implemented on the real plant, it had to be expanded and adapted to meet the requirements of a process computer. First of all the program for the state control system was overlaid by a diagnostic and supervisory level. Its main tasks are to monitor continuously the characteristic temperature curve (Equation (3)), handle overflow and underflow errors and perform broad plausibility checks on plant measurements. Before these values are further processed, the program compares them with predefined plausibility ranges. If any of the values lie outside the validity range, the responsible personnel are informed immediately and the control system is automatically switched to off-line operation. The state controller continuously generates the setpoint value for the steam flow. The setpoint value is used in an internal control loop in which an auxilliary controller (PI controller) is employed. The control parameters of this internal loop have been optimized for the existing characteristics of the control valve by a simulation program. The task of this controller is to maintain process conditions at the setpoint determined by the state controller. Furthermore, it was found that the dynamic behaviour of the column can be improved even more if mean values of the plant measurements over a short period are used as representative state values for the state control system. A period of 1 min was taken for calculating the mean values for all processed measurements, except in the case of the feed stream rate, where a period of 2 min was chosen. Figure 6 and Figure 7 are schematic representations of the control system. The program for the state control system was installed on a Siemens R30 process computer. It was connected to the real plant over the Teleperm M process control station. The total control system runs in real time. Measured values and setpoint values are transferred through a bus. Measured values and calculated setpoint values are stored for purposes of analysis and further optimization. This means that they are also available for graphical evaluation. The control system was implemented in two stages. First it was tested in off-line operation, receiving all plant measurements from the process control system and using them to calculate the setpoint of the manipulated variable. This setpoint was only used for purposes of comparison. The link between the state controller and the control valve was interrupted. The plant was still oper-
J. Proc. Cont. 1992, Vol2, No 4
183
State control system for an industrial distillation column: G. Fieg et al.
SO
1 r
state observer and controller
I
s
B X8
Figure 6
Control
system with state observer
1 PI auxiliary and state controller
ated with the conventional PID controller. These results have been described by Fieg et al.” and will not be more closely analysed in the context of this work. Instead the most important results of the second stage of the on-line testing will be described. The tests were carried out in two steps. In the first, the state control system was tested on-line for relatively short periods (2-6 h per test). The aim of these tests was to study the dynamic behaviour of the plant and the maintenance of the bottom concentration in semi steadystate conditions. This applied to all three state observers that were developed. Further, the performance of the transition between the individual observer models was examined and optimized. After these tests had been successfully concluded, the dynamic behaviour of the column at specific disturbances was investigated. Figure 8 shows a representative result of such tests. In a period of constant feed concentration (l&17% by weight) the feed rate (Figure 8~) was reduced from 4.8 to 4.5 m3 h-’ and then returned to 4.8 m3 h-‘. The heating steam rate followed this disturbance very quickly without overshoot and changed in a correspondingly proportional fashion. Figure 8b shows the paths followed by the manipulated variable in PID and state-space control. The first shows the change in the actual heating steam rate mD, the second the rate fiD estimated by the state control system. During the on-line phase both variables coincide. This is a measure of the performance of the PIauxiliary controller that was developed (Figure 7). It can be seen that it remains close to the setpoint given by the state controller. Similar tests of the dynamic behaviour of the column were also carried out in other concent-
184
J. Proc. Cont. 1992, Vol2, No 4
controller
1
l-l
Figure 7
Simplified
structure
of control
system
ration ranges. The results obtained were comparable with those shown in Figure 8. Figure 9 shows an example of dynamic plant behaviour when the plant was being brought up to a new production throughput. As can be seen from Figure 9a, the feed rate increased from 3 to 5.2 m3 h-’ within some 3-3.5 h. The feed concentration varied between 2&22% during this period. In Figure 9b the heating steam rate is plotted against time. Here too there was no overshoot and the manipulated variable (heating steam rate) soon reached the specified setpoint. Similarly good dynamic behaviour was also displayed by the column during reduction of the feed rate. Extreme fluctuation of the heating steam rate can be observed on the left of Figure 96. Within a short period the heating steam rate changed from 400 to 1000 kg h-‘. This period corresponds to the control concept with conventional PID controllers. The control variable (temperature) showed the same fluctuations as the heating steam. For this reason this representation was omitted. The state-control system was operating off-line. As already mentioned, the fluctuations in the manipulated variable are associated with the thermal coupling between the feed rate and the bottom product; the strong fluctuations
State control system for an industrial distillation column: G. Fieg et al.
al
al R
T,
2800
90
kg/h
Y
2500
86
8.01
LO -
F
XF
weight %. m’/h
40 leighlI % ‘ll 32
2200
_F
Xf
.
lb 1900
3.2.
16
J Xf
1.6.
8
1600
0
J
o.oJ
1300
bl
bl
1700
1700 kg/h
kg/h I
I
state
controtlcr
’,_ mo
1400
1400
ho. q
I_
1100
1100
VI* mo 800 800
0
2
4
6
8
10
12 h
14
time -
Figure 8 On-line test of state observer and controller in production column with induced disturbances (feed rate, state observer-model 2): a, disturbances versus time: b, manipulated variable wsus time
were rapidly eliminated by switching to the state control system. A similar case is shown in Figure 10. In Figure 1Oa feed rate and concentration are plotted against time, and in Figure ZOb heating-steam rate is plotted against time. Initially the state control system ran off-line for approximately 1.3 h. During a phase of decreasing heating steam fluctuations the state control system was switched online. A clear improvement in dynamic behaviour was observed. The fluctuations soon disappeared, and all in all the plant ran more smoothly. This changed after 3.6 h, when the control system with the PI controller was again switched on. The positive experience and results concerning the dynamic behaviour of the distillation column that were gained in the first on-line phase of state control system testing were subsequently checked in a long-term operational test (approximately 7 weeks). In the following section the results obtained during a 24 h period will be more closely analysed. They are given in Figure lla, and show marked feed rate fluctuations (F = 5 m3 h-’ + 3.8 + 4.7 m3 h-‘) overlaid by strong variations in concentration (19% by weight + 24% + 12% + 33% + 20%). These fluctuations correspond to transitions between the individual state observer models, and present an opportunity of assessing this aspect. In Figure 11 the heating steam rate is plotted against time during the observation period. The manipulated variable
500
200
0
2
L
6
8
h
10
Figure 9 Dynamic plant behavlour; teed rate changes up to desired production
follows a particularly smooth path, despite the disturbances. The performance of the transition between the observer models was good. In conjunction with Figures 9-11 it should be emphasized that every 4 h the product in the bottom of the column was sampled and its methanol content analysed using a gas chromatograph. The analysis of the bottom product showed fluctuation in the narrow range of concentration between 350 and 600 ppm. The product demands are therefore completely fulfilled.
Conclusions A systematic method for developing and implementing a state control system for the bottom-product concentration of an industrial distillation column was described. It covers the development of a mathematical model with system identification, the design and testing of a state observer and state controller, including the installation and startup of the complete control system. This procedure is characterized by the fact that practically all development steps were carried out by means of simulation calculations before the control system was implemented in the real plant. On the basis of the indivi-
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State control system for an industrial distillation column: G. Fieg et al. al state weight
%
m’/h
T, XF F
controller
I I I
I
xF
feed concentration 81 3L
F
4.0.
F Ii.
feed rate 5.1
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1.9
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'
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PI shtr ton~rdlcr
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1controller
bl
I
moS m0 300
0
0 0
12
3
time
4
2
4
6
8
10 12 14
5h6
--
16 18 20 h
time Figure 11
2k
-
Dynamic plant behaviour. Long-term operational test
Figure 10
On-line test of state observer and controller with heating steam rate fluctuations
m3 h-’ load) by using three linear observers of the same type, but with different parameters. dual tests carried out and the operational test, the operational experience gained can be summarized as follows.
References l
l
l
l
l
186
No adjustments had to be carried out to optimize the control system during the operational test; the control system functioned immediately. The plant exhibited particularly smooth dynamic behaviour during operation, even when extreme disturbances occurred. In this regard, the control system exhibited marked superiority over a conventional control system with PI controllers. No fluctuations resulted from disturbances. The transition between the individual observer models was good and posed no problems. As a result of the dynamic behaviour of the plant, it was possible to bring the methanol concentration of the waste-water closer to purity requirements (1000 ppm). Numerous examinations of the waste-water at 4 h intervals showed bottom concentrations between 350 and 600 ppm. These concentrations were 2-3 times lower than the permitted level of purity, whereas previously they were 516 times lower. It was possible to cope with the non-linearity of the plant within the ranges (8-35% methanol, 3-5
J. Proc. Cont. 1992, Vol2, No 4
1 2
Silberberger, F. PhD thesis, Universitat Stuttgart, 1978 Gilles, E. D. and Retzbach, B. IEEE Trans. Aufom. Control
3 4
9 10 11 12 13
Retzbach, B. Fortschr. Ber. VDZZ. 1986, Reihe 8 (126) Gilles, E. D., Retzbach, B. and Silberberger, F. in ‘Computer Applications to Chemical Engineering’ (Ed R. G. Squires and G. V. Reklaitis) ACS Symposium Series 1980 (124) 481 Eckelmann, W. Regel. Praxis 1980,22, (4), 120 Fieg, G., Wozny, G., Jeromin, L., KBhne, M. and Giilich, H. Vortrag auf dem GVC-Jahrestreffen der Verfahrensingenieure, Hannover, Germany, 1988 Marquardt, W. &em. Zng. Techn. 1989,61 (5), 362 Marquardt, W. in ‘Model Based Process Control Proceedings of the IFAC Workshop, Atlanta, USA, 13-14 June, 1988, (Eds T. J. McAvoy, Y. Arkum and E. Zafiriou) Pergamon Press, Oxford, 1989, 137 Krause, H. and Vahldieck, R. VDZ Ber. 1986,589, 73 Zeitz, M. Fortschr. Ber. VDZ Z. 1977,8 (27) Schuler, H. Fortschr. Ber. VDZ Z. 1982, Reihe 8 (52) Canavas, C. Forfschr. Ber. VDZ Z. 1988, Reihe 8 (171) MacGregor, J. F., Penlidis, A. and Ham&c, A. E. Polym.
14
Wozny, G., Witt, W. and Jeromin, L. Chem. Eng. Technol. 1987,
15 16 17 18
Skogestad, S. and Morari, M. AZChE J. 1987,33 (lo), 1620 Wozny, G. Habilitationsschrift, Universitgt Siegen, 1983 Luenberger, D. IEEE Trans. Mil. Electron. 1964, MIL-S, 74 Fieg, G., Wozny, G., Jeromin, L., Kiihne, M. and Giilich, H.
1983,28,628
5 6
I 8
Process Eng. 1984 2 (2 & 3), 179 10, 338
Chem. Eng. Technol. 1989, 12 339
State control system for an industrial distillation column: 19 20
Naphtali, L. M. and Sandholm, D. P. AIChEJ. 1971,17 (l), 148 Ferraris, B. G. AZChE J 1981,27 (l), 163
G. Fieg et al.
Appendix Matrices (Equations (4) and (5))
Nomenclature bottom product rate (kmol s-l) distillate rate (kmol s-‘) Feed rate (kmol s-l) feedback parameters (state observer) controller parameters coefficient (Equation (1)) coefficient (Equation (2)) heating steam rate (kmol ss’, kg s-‘) coefficient (Equation (3)) (“C stage-‘) reflux rate (kmol se’) location of the temperature front (stages) eigenvalues of state observer (s-l) eigenvalues of the control system (s-l) sidestream rate (kmol s-l) temperature (K) time constant (Equation (1)) (s) deviation of the manipulated variable (kmol se’) vapour rate inside the column (kmol s-l) liquid mole fraction (bottom product) (kmol kmol-‘) liquid mole fraction (distillate) (kmol kmol-‘) liquid mole fraction (feed) (kmol kmol-‘) deviation of measured variable (K) deviation of disturbance (kmol s-l)
Subscrlprs B D F
bottom process steam feed estimated value by the state observer
system matrix
manipulated input matrix
c=;II output
matrix
disturbance input matrix
Ll
q = g,
observer feedback matrix
k = [k,
k2]
x=
Av* ds
I I
controller feedback matrix state vector
u = AmD manipulated variable y =
AT
z =
A(F.+
measured variable + R)
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