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Theory and application of the modulating function method—III. application to industrial process, a well-stirred tank reactor
Computersthem. Engng, Vol. 17, No. 1, pp. 29-39, 1993 Printed in Gnat Britain. All rights reserved
0098.1354/93 $6.00 + 0.00 Copyright 0 1993 Pergamon Press Ltd
THEORY AND APPLICATION OF THE MODULATING FUNCTION METHOD-III. APPLICATION TO INDUSTRIAL PROCESS, A WELL-STIRRED TANK REACTOR H. A.
PREISIG’t
and D. W. T.
RIPPING
‘School of Chemical Engineering and Industrial Chemistry, University of New South Wales, Kensington (Sydney), NSW 2033, Australia ‘Technisch Chemisches Laboratorium, Swiss Federal Institute of Technology CH-8092 Zurich, Switzerland (Received
14 June 1988; jnal
(ETH)
revision received 5 May 1992; received for publication 21 May 1992)
Abstract-A dynamic model describing the energy dissipation in a poorly defined, industrial, well-stirred tank reactor is identified. A successive refinement approach is presented in which an initial simple model is refined in three stages: (i) the modulating function method is utilized for estimating the heat transfer parameters locally as a function of time; (ii) the parameters, which, because of the modelling errors, change with changing operating conditions, are graphically correlated with the operating conditions; and (iii) the resulting nonlinear model is refined by introducing additional dynamic elements. Validation of the model was done by comparing the predicted steady-state heat losses with other experimental data.
recipe which prescribes the sequential operation of the plant to produce the desired product with high concentration, selectivity and yield. This open-loop control approach is certainly not optimal because any disturbance not observed and compensated for will potentially cause a deviation from the desired path. Feedback compensation is only provided at a low level and, to a limited extent, in the implemented production recipe, which is usually the result of extensive laboratory studies. Research has therefore concentrated on using available process information for the reconstruction of the composition in the reactor so that a controller can be designed to adjust for disturbances on this level (Wallman and Foss, 19891; Weber and Brosilow, 1972). This problem is called the observer problem, and is well known in the literature (Kalman and Bucy, 1961; Luenberger, 1971). The key component of an observer algorithm is a process model, which provides the connection between the measurable quantities and the quantities that shall be reconstructed. For a chemical production unit this model describes the relation between the measurable process quantities and the composition of the product, which implies that it includes a description of the physical system as well as the chemical system (Preisig, 1988, 1989a, 1990; De Valliere, 1989). As the process model is the main component of the observer, effects of modelling errors are of major concern. A study of the effect of modelling errors on the observation of the composition in a batch reactor was described in Preisig (1989b).
1. INTRODUClJON
third paper in this series is devoted to an industrial application utilizing the method described in Part I (Preisig and Rippin, 1993a) using the software based on the theory described in Part II (Preisig The
and
Rippin,
approach
1993b)
and
the
for identifying
successive
poorly-defined
refinement systems
as
used as an example in Part II. The following section of this paper describes the background and purpose for which the model was developed. A brief description of the plant is followed by a detailed description of a simple process model, which stimulates the dynamic energy (heat) dissipation in the stirred tank reactor. Finally, the accuracy with which the model reproduces the process is analyzed based on alternative data sets. The paper closes with some conclusions. 2. PROJRa
BACKGROUND
In many chemical plants, particularly batch plants, major operational problems arise because no direct information about the concentration of the product is available. The lack of information on this level is caused by the lack of component-selective sensors. As one of the consequences, process control on the composition level is open loop and the process units or plants are usually controlled following a tTo whom all correspondence
should be addressed. 29
30
H. A,
hEISlG
and D. W. T.
The study presented in this paper reports on the identification of the physical part of the overall process model for a stirred tank reactor, namely the heat distribution. The objective in this project was to generate procedures to observe the heat generation term on-line in an industrial pilot plant. From this information a second observer would then reconstruct the composition using a kinetic model of the chemistry. The data were collected in industry and analyzed at the university to provide a sample case study demonstrating the feasibility of reconstructing the heat generation term under unfavorable industrial conditions.
kPPlN
reflux and distillation. Filters and crystallizers are available for product treatment. Subject of our investigation was the central part, the tank. Its heating cooling system consists of an inner recycle loop with a high flowrate of the heating/ cooling medium maintained by a circulation pump. This inner cycle is connected to a multiloop outer heating cooling system with four streams of different temperatures ranging from a brine-cooled lowtemperature cycle to a high-temperature cycle heated with superheated stream. The whole plant is connected to a process computer which supervises the operations. Data acquisition and control are done with a constant sampling interval of 2 s.
3.THEPLANT 4. PROCESS
The topology of the pilot plant being investigated, shown in Fig. 1, is typical for the batch process industry. The heart of the plant is a jacketed wellstirred tank. The jacket is split into two sections, the bottom part and the cylindrical part; an arrangement which allows safe handling of small volumes as well as efficient heating and cooling of large quantities. The whole unit is modular in the sense that the stirrer can be exchanged allowing for different stirrer types. Also different feed arrangements are possible and a rectifying column mounted on the lid allows for
IDENTIFICATION
4.1. Process model A close look at the system reveals several facts which ought to be borne in mind when modelling the system: (i) the lid is a massive construction because it serves as the base for mounting all the equipment such as the stirrer and distillation column onto the tank; (ii) the walls of the jacket are thick and massive because the jacket operates under elevated pressure; (iii) the jacket is insulated externally but the lid and all the mounting devices such as the flanges are not
loT
Q
u-=T
F
Temperature Weight Fbr Speed of Rotatbn Torque
Fig. 1. Pilot plant scheme.
Theory and application of modulating function method-III insulated; (iv) the jacket is divided into two sections; and (v) temperatures are measured in the stream entering the jacket before it splits into the feeds for the two sections and after joining again the outputs of the two jacket sections, the tank contents, the gas phase above the tank contents, on the lid and in the room. The model of the energy distribution in the system is based on energy balances drawn up over domains in the 3-D space. The boundaries of the accessible control volume defining the elements included in the system are determined by the location of the installed measurements. From this point of view, the system is roughly dehned by the surface of the tank cutting through the inner heating/cooling recycle streams at the location of the two tempeature measurements (one before and one after the jacket) indicated by the dotted line in Fig. 1. Although the element of interest is the liquid contents of the tank and all the other elements within the control volume including the wall between the tank and the jacket, the gas phase, the lid and its associated items have to be included in the model. Based on experiences with the plant and the identification of several different models with different degrees of detail, a very simple model proved to be the best candidate. It lumps several of the fast components of the system together, particularly in the domain of the jacket which is modelled as a single lumped system. Thus, the “fast” components of the system are modelled crudely. On the other hand, more effort has been expended on modelling the slow components, which alfect the steady-state description. One of these components is the heat exchange between the contents and the room through the inside gas phase and the lid. These two elements, the lid and the gas phase, are therefore part of the model. Overall the model splits the system into four lumps assuming that energy is exchanged only between adjacent lumps. The model has thus the structure of a chain of interconnected first-order systems. At one end, the
where
3 c L R y; y,
:: :: :: :: :: ::
Jacket Contents Lid Room enthalpy enthalpy
; 1, ,I?
in out
c CF
energy flow into the chain is controlled by the heating/ cooling system. The jacket communicates with the contents, which connect through the gas phase and the lid to the room at the other end of the chain. The room itself is modelled as an infinite energy reservoir. Mathematically, the mode1 assumes the form: CjJ’,= --hjc(Yj -Yc) +#(Yi*YoT IQ cc.Y== &(Yj -Yc) -&,(Yc
-YE),
cs,‘, = &,(Y, -Ye) -h,(Y,-Y,), CIYI= h,(Y, - n) - h&Y, - Y,),
(El)
where c, hs,s*
HTP 4 ri Y, io,j.c,g,l.r
:: heat capacity of lump s (kJK_‘) transfer parameter (HTP) between lump s, and s, (kW K-l), :: heat transfer coefficient multiplied by the heat transfer area, :: heat input in the jacket (kW), :: volumetric flowrate of the heating/cooling medium through the jacket (m’s -I), :: measured temperature of lump s, gas :: jacket in, jacket out, jacket, contents, phase, lid, room. :: heat
Some of the parameters were evaluated from the technical drawings including the mass of the vessel, the volume of the gas phase and the jacket contents. The latter was verified by measuring the volume directly. The mass of the baffles, the lid and the stirrer were measured by weighing. All physical properties of the equipment parts were assumed to be temperature independent. For the gas phase this is quite a crude assumption, which however, because of its small mass, does not affect the accuracy of the description to speak of. In the time scale of interest, the gas phase is dynamically of no importance. It only acts as a resistance to the transfer of energy between the contents and the lid.
:: Heat capacity := pVc, :: Heat transfer parameter between lump 11 and 1s :: Density :: Volume :: Specific heat capacity / mass
Fig. 2. The structure of the initial simple model. CACE 17/1-C
31
32
H. A.
t’REISIG
and D. W. T.
4.2. Approach to model ident@cation The structure of the system makes it obvious that a very accurate mechanistic description cannot be attempted, at least, not in one step. For this reason, a successive refinement approach was chosen building stepwise in a more complex and accurate description starting from the simple model presented in (El). The proposed approach is based on the observation that any errors in the model structure will map into the estimated parameters. Assuming that a method is available for the estimation of the parameters of the simple model at any point in time using only data about this point in time, the parameters of an erroneous model will not be constant but will change with changing operating conditions and since the operating conditions change with time the parameters will fluctuate with time too. Since the modelling errors introduce systematic errors, relations between the parameters of the original simple model and the operating conditions, or the changing states describing the system behaviour, can be identified. Parameters that are associated with the dynamic state equations are related with the particular state variable, whilst parameters associated with transfer laws, are corrected with the states of the two subsystems connected by the transfer of extensive quantity. These mostly empirical relations then serve to suggest extensions of the existing model leading to the introduction of nonlinearities. The procedure could be repeated in that the parameters (all or only some of them) of the extended model could be identified using the experimental data and the modulating function method again, which also applies to nonlinear models within the limits analyzed in Preisig and Rippin (1993a). Again, errors could possibly be reduced by correlating the new set of parameters with
the changing operating conditions. Theoretically, this procedure could be repeated with changing models and their respective parameters until a model is obtained that describes the process sufficiently accurately for the application. This approach requires point estimates, meaning estimates local in time, which automatically implies that differential information about the process is required. When a physical rather than a simulated system is treated noise will be present in the measurements, direct differentiation of the measured signals
RIPPIN
is not feasible and prefiltering is required. In the past, research workers and engineers used predominately Kahnan filtering combined with rectangular and exponential data-windowing to solve this problem. Another popular method was straightforward minimal-sumof-squares filtering combined with data-windowing to estimate the initial conditions together with the parameters. Kalman filtering was also the first choice of the authors in this case. However, after a year of intensive work no well-based argument could be found for choosing the structure and values of the two variance-covariance matrices which directly determine the distribution to the states or the parameters of the errors caused by the inaccuracy of the model. As a consequence, an alternative method was sought. Shinbrot’s modulating function approach was designed exactly for solving problems of this kind. It also solves another problem namely that of eliminating the unknown initial conditions for every interval estimate. The modulating function method, which has been introduced in the first part of this series (Preisig and Rippin, 1993a), also chooses a window of data; but since the initial conditions are eliminated, the dimension of the identification problem is reduced by the number of initial conditions, which is identical to the number of states. The spline-type modulating function, Male&sky’s extension of Shinbrot’s method, proved to be a very powerful method for solving this kind of problems. Applying the modulating function method to the model (see the Appendix), the following matrix equation is obtained which is discrete in time because. of the discrete sliding of the modulating functions. ARk)#k)
The vector and matrices
=
Wkll.
in this equation
P=(~)~=V$c, h,, h,, &I,
032) are
CD11
The parameters are obtained as a function of time from equation (E2) by solving for the parameter vector p for each set of data. Consequently, the parameters are obtained as a discrete signal from the identification stage with the same discrete time index as the modulated signals.
33
Theory and application of modulating function method-III 100 90
I-
jacket(average) _________ contents -
0
0.2
0.4
0.6
0.8
1
1.2
Time [s]
room
1.4
1.6
1.8
2! x104
Fig. 3. Stepwise heating and cooling.
4.3. Experiment The experiment was a combination of high- and low-frequency excitation covering a range of reasonable operating conditions. Several experiments were done. The most informative ones excited the jacket with a stair input increasing the temperature in the jacket stepwise by 10°C starting at a room temperature of about 20°C up to about 8O“C. Figure 3 shows the measured temperatures after correcting for the offsets in the measurements. The offsets were determined in a separate experiment and verified before the 6rst temperature step in this experiment. 4.4. Model rejnement The filtering properites were chosen to match the dominating time constant for the dynamic process of interest. Since the heat transfer through the wall dominates the dynamics of the overall heat distribution the major time constant of this subprocess was estimated and using as the characteristic time. A value of about 50 s was found to be feasible. The
tThe order of the modulating function v is usually chosen one bigger than the order of the model hecause the vth derivative of the modulating function, the stem function, is a set of Dirac delta functions. The stem functions, thus, use only the signal values that coincide with the Dkac impulses; all other information that lies in between is not utilized, which is the resson the stem function is usually not used (see Part I, !Iection 5.3).
choice is not critical. Variations of + 10 s do not affect the results to speak of. The order of the modulating function chosen is 2, which is one bigger than the highest order in the model equations.? Again, the choice of order does not affect the results visibly. The chioce of how much the modulating functions overlap or in other words the choice of the shift time AT has been discussed in Preisig and Rippin (1993a). A value of T/AT E [5. . . 101is a good rule of thumb. It should be noted that the shifting of the modulating function does not imply discrete processing of the measured data but only means that the output of the modulating filtering operation is discrete with a sampling time of AT. The results, shown in Fig. 4, have some interesting characteristics: (i) a trend between the HTP and the operating conditions is observed. With increasing temperature the HTP increase and vice versa; (ii) large, biased errors can be observed at the beginning of each step; and (iii) in the second phase, the cooling phase of the experiment, large errors are observed in the HTP. 4.4.1. Large errors in the cooling phase. The large errors in the cooling phase are due to a decreasing signal/noise ratio. As the process of heating reverts to cooling, the temperature profiles of the different elements cross each other. Since the driving force for the heat transfer is the temperature difference, no information remains in the temperature difference signal and the process cannot be identified. In other words, the system is not persistently excited (Astrom
H. A.
34
h@lslG and
D. W. T.
RIPP~N
1
-
jacket-contents ....... contents-gas phase ............. ize:-hd ..........
0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 C
D.2
a 1.4
0.6
0.8
1.2
1.4
1.6
1.8
Time Is]
2 x104
Fig. 4. The heat transfer parameters (HTP) as a function of time.
and Bohlin, 1966) and since the level of persistent excitation can only be estimated the interval during which the subprocesses are not identifiable can also only be estimated. The length of this interval depends on the noise level and the dynamics of the decrease
0.9
-
heating cooling
..._....
mdel
. ........ ...
and increase of the temperature difference. The large errors in the parameter estimates that are observed during the later part of the cooling phase are therefore not a weakness of the identification procedure but a consequence of the experimental conditions.
0.8 0.7
i
0.6 11.
D TetnpemmeConW Fig. 5. HTF’jacket--contents
80
[Cl
correlated with the temperature in the contents; empirical linear heat transfer parameter model.
Theory and application of modulatiug function method--III
35 I
Temperamre Gas Phase [Cl Fig. 6. HTF’ contents--gas
phase correlated with the temperature in the gas phase.; empirical exponential heat transfer parameter model.
4.4.2. Nonlinearities. The model parameters were identilkd using small sets of data. The data window, given by the length of the modulating function, is discretely shifted over the signals generating the discrete modulated signals. From these modulated
signals the parameters are calculated allowing them to slowly change with time and operating conditions. The changes, or trends, in the parameters are then used to extend the first simple model. The heat transfer parameters are graphically correlated with
-1
-2
-3
-4
-5 .... ... . ..,
-6
heating
mling ._______mdel
-7 20
30
40
50
60
70
Temperamre Gas Phase [Cl Fig. 7. HTP gas phase-lid
correlated with the temperature in the gas phase; empirical exponential heat transfer parameter model.
36
H.
A.
PREISIG
and D. W. T.
RIPPIN
0
I
. .. . -1 -
3
_________
3.1
3.2
hesting cooling model
3.3
3.4
3.5
3.6
3.7
3.8
3.9
ln( Temperature Difference (Lid-Room) [K] ) Fig. 8. HlT lid-room
correlated with the temperature difference between the. lid and the room; empirical power law heat transfer parameter model.
the operating conditions neglecting the biases and the intervals where the processes are not identifiable during the cooling phase. Figure 5 shows the HTP for the heat exchange between the jacket and the contents correlated with the temperature of the contents. The straight line represents the empirical function chosen for describing the correlation between the two quantities. Figures 6 and 7 show correlations between the HTP contents-gas phase and gas phase-lid, respectively. In both cases, an exponential relation seems to describe the behaviour of the HPT’s well. The actual heat transfer process is a mixture of heat conduction and convective evaporation and condensation. The conduction probably dominates at relatively low temperatures. As the temperature increases, the heat transfer due to evaporation and condensation increases until it completely dominates at boiling temperature. The last plot, Fig. 8, shows the correlation of the HTP for the heat exchange between the lid surface and the room. The empirical power law of natural convection was assumed to apply with a temperature difference raised to the poweri of I .25. As can be seen from the plot, this assumption fits the experimental data well. $This co&Cent includes the temperature dependency of the heat transfer constant.
4.4.3. Dynamic errors. The biased estimates obtained at the beginning of each step indicate that the model is not capable of describing the fast parts of the experiment accurately. Several problems can be identified. Firstly, the model describes the capacities in the area of the jacket with a single lump. Secondly, the model assumes uniform conditions in the jacket. Thirdly, heat transfer was assumed to occur between adjacent lumps only. Fourthly, the measurement devices are assumed to have no time lag. Using tbe nonlinear extensions identified above and comparing the temperature of the contents measured with the temperature reconstructed by the model, the problems with the dynamics become even more obvious. Also, even though special attention was paid to calibration of the temperature measurement devices, some calibration errors still remained as the difference does not approach zero at the end of the experiment. The calibration problem could not be corrected given the experimental set up (Fig. 9). The experiment presented in this paper was the best one obtained from this plant. The dynamic effects, however, were studied in separate experiments. The first one focused on the slow temperature measurements, mainly the sensor in the contents which, for mechanical reasons, is built into a solid support with a correspondingly high thermal capacity. Its time lag was estimated to fall within the range 20-60 s, dependent on the flow conditions.
37
Theory and application of modulating function method-III 1.5
1
0.5 ,
-Ieumulatlve error
-0.5
-1
0
200
400
600
800
1000
1200
1400
1600
1800
:
Cl
Fig. 9. Difference between the measured and the predicted temperature in the contents without dynamic corrections. The second experiment involved a small experimental setup, which consists of two tanks cut into a block Styrofoam separated by a wall of the same quality as the one in the reactor. Dynamic experiments were performed changing the temperature by
injecting hot water into one of the tanks which was half filled with water at room temperature. These experiments, discussed in more detail in Preisig (1984) and Koster (198 l), showed that a second-order model for the wall results in a dramatic improvement of the
time lag -0.5 -
6
c1
-1
0
-___-___...
0.2
0.4
0.6
1
0.8
Time [s]
1.2
1.4
ZS 40 s
1.6
1.8
2 x104
Fig. 10. Difference between the measured and the predicted temperature in the contents, with dynamic
wrreetions.
H.
38
A.
hEISIG
and D. W. T.
RPPIN
Table 1 COMtitiOtM
Steadv-state wints 43.5 37 32 22
53.5 44 38 22
63.5 52 49 22
73.5 63 61 22
83.5 18 77 22
Measured heat losses Estimated heat losses
0.15 0.17
0.22 0.27
0.53 0.40
0.59 0.59
0.89 0.87
dynamic description. This approach has therefore been adopted in this study. The fluid-film resistances on the two sides of the wall are assumed to be of equal magnitude, knowing the resistance introduced by the wall because of known physical properties and geometry. Figure 10 shows the comparison between the measured temperature and the model responses with different time lags for the temperature sensor in the contents. Including the sensor dynamics improves the results significantly. The error is reduced by a factor of two and remains over the whole experiment within the limits of kO.6 K. 4.4.4. Steady state errors. Errors in the steady-state description would, to a certain extent, accumulate in an observer scheme. Therefore, the accuracy of the model in describing the steady-state behaviour of the plant, mainly the heat losses, is important. For the description of the heat losses through the lid, a number of ad hoc assumptions have been made and must be checked by comparing the predicted steady-state heat losses and the measured heat losses. The steady-state temperatures in the gas phase and on the surface of the lid can be calculated from the model by zeroing the accumulation terms on the left-hand side and substituting the models for the HTP parameters resulting in two nonlinear equations in the gas and lid temperature. The two equations are:
=fl exp&y,l(~,
-n)
- ~I,IY, -Y~I~~~~(Y~ -Y,) = 0, (E3)
where ai,jv&j
::
Units
Temperature contents Estimated temperature gas phase Estimated temperature lid Room tmmxrature
the parameters in the exponential relations describing the heat transfer between lump i, j.
These equations are readily solved for the steady-state temperatures using a Newton-Raphson iteration scheme. Table 1 shows the comparison of predicted steady-state heat losses and values measured in an experiment that had been performed under very similar operating conditions about a year earlier. The accuracy with which the model describes the steady-state heat losses is extremely good, only the value at y. = 63.5”C deviates significantly.
“C DC “C “C
Lastly, the steady-state analysis can be used to estimate the errors between the dynamic description and the steady-state description. Figure 11 shows a comparison of the predicted steady-state temperature differences and the measured values. The differences between the measured and the steady-state predictions are quite small during the heating phase. However, the differences are much bigger during the cooling phase. These graphs show the magnitude of the dynamic errors that have to be expected. This more sluggish behaviour of the process to cooling is confirmed by practitioners operating industrial plants. The quality of the identified model can also been judged by analyzing the accuracy of the simulated data using the model outside the range of operating conditions used for the identification of the model. The temperature differences between the contents, the gas phase and the lid decrease rapidly with increasing system temperature, as one would expect. 5.
SIGNIFICANCE
AND CONCLUSIONS
The pilot plant for which a dynamic description of the heat dissipation was sought had all the attributes that are typical for an industrial plant: the heat capacity of the construction elements are not negligible and of a distributed nature, the measurements are not very accurate and the heat losses are significant depending on the operating conditions. These are all reasons for which it is literally impossible to formulate and identify a very detailed model of the process. In reviewing the results gained in this investigation the authors arrived at the opinion that the approach of successively refining a simple model offers an attrao tive solution to the problem of identifying a dynamic model for a badly defined system. The modulating function method-in particular Maletinsky’s splinetype modulating functions--proved to be a powerful tool for evaluating the parameters as the process proceeds. Correlating the parameter variations induced by modelling errors with the changing state variables results in nonlinear extensions of the original simple model. At the present time, graphical analysis appears to be the most appropriate way of determining these extensions since it enables the analyst to qualitatively weigh the different sections of the experiment.
Theory and application of modulating function method-III 50
39
I
40-
contents-gas phase
heating cooling
20
30
40
50
60
70
80
90
100
Temperature Contents [Cl Fig. Il. Measured
vs calculated
Considerable improvement in model accuracy can be achieved by simple dynamic extensions that satisfy steady-state conditions. The steady-state heat losses predicted by the extended model compare excellently with experimental data obtained from the plant. REFRmNCES
Astriirn K. J. and T. Bohlin, Numerical identification of linear dynamic systems from normal operating records. IFAC Symp. Theory of Self-Adaptive Control Systems, Teddington. England in Theory of Self-Aakptive Control Systonr (Edited by P. H. Hammond). Plenum Press, New York (1966). De Valli&e P. R. L., State and parameter estimation in batch reactors for the purpose of inferring on-line the rate of heat production. Ph.D. Thesis, Swiss Federal Institute of Technology, Diss ETH Nr 8847, Zurich (1989). Kalman R. E. and R. S. Bucy, New results in filtering and prediction theory. J. Basic Engng Man, (1961). Koster U., Identi8kation einfacher Modelle filr einen verteilten W&medurchgang; Diploma Thesis No. 3900, Swiss Federal Institute of Technology (ETH), Tech. Chem. Lab. (1981). Luenberger D. G., An introduction to observers. IEEE Trans AC Control AC-16, (1971). Preisig H. A., On the identification of structurally simple dynamical models for the energy distribution in stirredtank reactor equipment. Ph.D. Thesis, Swiss Federal Institute of Technology, Diss ETH Nr 7616, Zurich (1984). Preisig H. A., The use of differential information for batch reactor control. ACC 88, Atlanta (1988). Preisig H. A., On-line observation of the composition in
steady-state
temperature
differences.
non-isothermal, batch reactors with nonlinear reactions. ACC 89, Pittsburgh (1989a). Preisig H. A., Effect of modelling errors on the observation of concentrations in batch reactors. AIChE Annual Meeting, 1989, Paper 167Bq, San Francisco (1989b). Preisig H. A., Application of observer using differential information to bench-scale reactor. ACC 90, San Diego (1990). Preisig H. A. and D. W. T. Rippin, Theory and application of the modulating function method-I. Review and theory of the method and theory of the spline-type modulating functions. Computers them. Engng 17, l-16 (1993a). Preisig H. A. and D. W. T. Rippin, Theory and application of the modulating function method-II. Algebraic representation of Maletinsky’s spline-type modulating functions. Computers &em. Engng 17, 17-28 (1993b). Wallman P. H. and A. S. Foss, Experiences with dynamic estimators for fixed-bed reactors. Ind. Engng Chem. Funaiun. 20, (1981). Weber R. and C. Brosilow, The use of secondary measurements to improve control. AIChE JI (1972).
APPENDIX Modulation of the model equations (El) (multiplication of the set of equations with the modulating integration by parts) results in: qs=
-h,(jl,-k)+q(jW,.
function and
p),
e,~==~(q-~~)-h,(~~-_3r), c‘~~=Eh,(~=-~~)-4(4-A), c,? =h~(~*-a)-ht(a-~~). (E4) Rewriting in matrix form and substituting the definitions (Dl), (D2) (D3) and (D4) equation (E2) is obtained.
0098.1354/93 $6.00 + 0.00 Copyright 0 1993 Pergamon Press Ltd
THEORY AND APPLICATION OF THE MODULATING FUNCTION METHOD-III. APPLICATION TO INDUSTRIAL PROCESS, A WELL-STIRRED TANK REACTOR H. A.
PREISIG’t
and D. W. T.
RIPPING
‘School of Chemical Engineering and Industrial Chemistry, University of New South Wales, Kensington (Sydney), NSW 2033, Australia ‘Technisch Chemisches Laboratorium, Swiss Federal Institute of Technology CH-8092 Zurich, Switzerland (Received
14 June 1988; jnal
(ETH)
revision received 5 May 1992; received for publication 21 May 1992)
Abstract-A dynamic model describing the energy dissipation in a poorly defined, industrial, well-stirred tank reactor is identified. A successive refinement approach is presented in which an initial simple model is refined in three stages: (i) the modulating function method is utilized for estimating the heat transfer parameters locally as a function of time; (ii) the parameters, which, because of the modelling errors, change with changing operating conditions, are graphically correlated with the operating conditions; and (iii) the resulting nonlinear model is refined by introducing additional dynamic elements. Validation of the model was done by comparing the predicted steady-state heat losses with other experimental data.
recipe which prescribes the sequential operation of the plant to produce the desired product with high concentration, selectivity and yield. This open-loop control approach is certainly not optimal because any disturbance not observed and compensated for will potentially cause a deviation from the desired path. Feedback compensation is only provided at a low level and, to a limited extent, in the implemented production recipe, which is usually the result of extensive laboratory studies. Research has therefore concentrated on using available process information for the reconstruction of the composition in the reactor so that a controller can be designed to adjust for disturbances on this level (Wallman and Foss, 19891; Weber and Brosilow, 1972). This problem is called the observer problem, and is well known in the literature (Kalman and Bucy, 1961; Luenberger, 1971). The key component of an observer algorithm is a process model, which provides the connection between the measurable quantities and the quantities that shall be reconstructed. For a chemical production unit this model describes the relation between the measurable process quantities and the composition of the product, which implies that it includes a description of the physical system as well as the chemical system (Preisig, 1988, 1989a, 1990; De Valliere, 1989). As the process model is the main component of the observer, effects of modelling errors are of major concern. A study of the effect of modelling errors on the observation of the composition in a batch reactor was described in Preisig (1989b).
1. INTRODUClJON
third paper in this series is devoted to an industrial application utilizing the method described in Part I (Preisig and Rippin, 1993a) using the software based on the theory described in Part II (Preisig The
and
Rippin,
approach
1993b)
and
the
for identifying
successive
poorly-defined
refinement systems
as
used as an example in Part II. The following section of this paper describes the background and purpose for which the model was developed. A brief description of the plant is followed by a detailed description of a simple process model, which stimulates the dynamic energy (heat) dissipation in the stirred tank reactor. Finally, the accuracy with which the model reproduces the process is analyzed based on alternative data sets. The paper closes with some conclusions. 2. PROJRa
BACKGROUND
In many chemical plants, particularly batch plants, major operational problems arise because no direct information about the concentration of the product is available. The lack of information on this level is caused by the lack of component-selective sensors. As one of the consequences, process control on the composition level is open loop and the process units or plants are usually controlled following a tTo whom all correspondence
should be addressed. 29
30
H. A,
hEISlG
and D. W. T.
The study presented in this paper reports on the identification of the physical part of the overall process model for a stirred tank reactor, namely the heat distribution. The objective in this project was to generate procedures to observe the heat generation term on-line in an industrial pilot plant. From this information a second observer would then reconstruct the composition using a kinetic model of the chemistry. The data were collected in industry and analyzed at the university to provide a sample case study demonstrating the feasibility of reconstructing the heat generation term under unfavorable industrial conditions.
kPPlN
reflux and distillation. Filters and crystallizers are available for product treatment. Subject of our investigation was the central part, the tank. Its heating cooling system consists of an inner recycle loop with a high flowrate of the heating/ cooling medium maintained by a circulation pump. This inner cycle is connected to a multiloop outer heating cooling system with four streams of different temperatures ranging from a brine-cooled lowtemperature cycle to a high-temperature cycle heated with superheated stream. The whole plant is connected to a process computer which supervises the operations. Data acquisition and control are done with a constant sampling interval of 2 s.
3.THEPLANT 4. PROCESS
The topology of the pilot plant being investigated, shown in Fig. 1, is typical for the batch process industry. The heart of the plant is a jacketed wellstirred tank. The jacket is split into two sections, the bottom part and the cylindrical part; an arrangement which allows safe handling of small volumes as well as efficient heating and cooling of large quantities. The whole unit is modular in the sense that the stirrer can be exchanged allowing for different stirrer types. Also different feed arrangements are possible and a rectifying column mounted on the lid allows for
IDENTIFICATION
4.1. Process model A close look at the system reveals several facts which ought to be borne in mind when modelling the system: (i) the lid is a massive construction because it serves as the base for mounting all the equipment such as the stirrer and distillation column onto the tank; (ii) the walls of the jacket are thick and massive because the jacket operates under elevated pressure; (iii) the jacket is insulated externally but the lid and all the mounting devices such as the flanges are not
loT
Q
u-=T
F
Temperature Weight Fbr Speed of Rotatbn Torque
Fig. 1. Pilot plant scheme.
Theory and application of modulating function method-III insulated; (iv) the jacket is divided into two sections; and (v) temperatures are measured in the stream entering the jacket before it splits into the feeds for the two sections and after joining again the outputs of the two jacket sections, the tank contents, the gas phase above the tank contents, on the lid and in the room. The model of the energy distribution in the system is based on energy balances drawn up over domains in the 3-D space. The boundaries of the accessible control volume defining the elements included in the system are determined by the location of the installed measurements. From this point of view, the system is roughly dehned by the surface of the tank cutting through the inner heating/cooling recycle streams at the location of the two tempeature measurements (one before and one after the jacket) indicated by the dotted line in Fig. 1. Although the element of interest is the liquid contents of the tank and all the other elements within the control volume including the wall between the tank and the jacket, the gas phase, the lid and its associated items have to be included in the model. Based on experiences with the plant and the identification of several different models with different degrees of detail, a very simple model proved to be the best candidate. It lumps several of the fast components of the system together, particularly in the domain of the jacket which is modelled as a single lumped system. Thus, the “fast” components of the system are modelled crudely. On the other hand, more effort has been expended on modelling the slow components, which alfect the steady-state description. One of these components is the heat exchange between the contents and the room through the inside gas phase and the lid. These two elements, the lid and the gas phase, are therefore part of the model. Overall the model splits the system into four lumps assuming that energy is exchanged only between adjacent lumps. The model has thus the structure of a chain of interconnected first-order systems. At one end, the
where
3 c L R y; y,
:: :: :: :: :: ::
Jacket Contents Lid Room enthalpy enthalpy
; 1, ,I?
in out
c CF
energy flow into the chain is controlled by the heating/ cooling system. The jacket communicates with the contents, which connect through the gas phase and the lid to the room at the other end of the chain. The room itself is modelled as an infinite energy reservoir. Mathematically, the mode1 assumes the form: CjJ’,= --hjc(Yj -Yc) +#(Yi*YoT IQ cc.Y== &(Yj -Yc) -&,(Yc
-YE),
cs,‘, = &,(Y, -Ye) -h,(Y,-Y,), CIYI= h,(Y, - n) - h&Y, - Y,),
(El)
where c, hs,s*
HTP 4 ri Y, io,j.c,g,l.r
:: heat capacity of lump s (kJK_‘) transfer parameter (HTP) between lump s, and s, (kW K-l), :: heat transfer coefficient multiplied by the heat transfer area, :: heat input in the jacket (kW), :: volumetric flowrate of the heating/cooling medium through the jacket (m’s -I), :: measured temperature of lump s, gas :: jacket in, jacket out, jacket, contents, phase, lid, room. :: heat
Some of the parameters were evaluated from the technical drawings including the mass of the vessel, the volume of the gas phase and the jacket contents. The latter was verified by measuring the volume directly. The mass of the baffles, the lid and the stirrer were measured by weighing. All physical properties of the equipment parts were assumed to be temperature independent. For the gas phase this is quite a crude assumption, which however, because of its small mass, does not affect the accuracy of the description to speak of. In the time scale of interest, the gas phase is dynamically of no importance. It only acts as a resistance to the transfer of energy between the contents and the lid.
:: Heat capacity := pVc, :: Heat transfer parameter between lump 11 and 1s :: Density :: Volume :: Specific heat capacity / mass
Fig. 2. The structure of the initial simple model. CACE 17/1-C
31
32
H. A.
t’REISIG
and D. W. T.
4.2. Approach to model ident@cation The structure of the system makes it obvious that a very accurate mechanistic description cannot be attempted, at least, not in one step. For this reason, a successive refinement approach was chosen building stepwise in a more complex and accurate description starting from the simple model presented in (El). The proposed approach is based on the observation that any errors in the model structure will map into the estimated parameters. Assuming that a method is available for the estimation of the parameters of the simple model at any point in time using only data about this point in time, the parameters of an erroneous model will not be constant but will change with changing operating conditions and since the operating conditions change with time the parameters will fluctuate with time too. Since the modelling errors introduce systematic errors, relations between the parameters of the original simple model and the operating conditions, or the changing states describing the system behaviour, can be identified. Parameters that are associated with the dynamic state equations are related with the particular state variable, whilst parameters associated with transfer laws, are corrected with the states of the two subsystems connected by the transfer of extensive quantity. These mostly empirical relations then serve to suggest extensions of the existing model leading to the introduction of nonlinearities. The procedure could be repeated in that the parameters (all or only some of them) of the extended model could be identified using the experimental data and the modulating function method again, which also applies to nonlinear models within the limits analyzed in Preisig and Rippin (1993a). Again, errors could possibly be reduced by correlating the new set of parameters with
the changing operating conditions. Theoretically, this procedure could be repeated with changing models and their respective parameters until a model is obtained that describes the process sufficiently accurately for the application. This approach requires point estimates, meaning estimates local in time, which automatically implies that differential information about the process is required. When a physical rather than a simulated system is treated noise will be present in the measurements, direct differentiation of the measured signals
RIPPIN
is not feasible and prefiltering is required. In the past, research workers and engineers used predominately Kahnan filtering combined with rectangular and exponential data-windowing to solve this problem. Another popular method was straightforward minimal-sumof-squares filtering combined with data-windowing to estimate the initial conditions together with the parameters. Kalman filtering was also the first choice of the authors in this case. However, after a year of intensive work no well-based argument could be found for choosing the structure and values of the two variance-covariance matrices which directly determine the distribution to the states or the parameters of the errors caused by the inaccuracy of the model. As a consequence, an alternative method was sought. Shinbrot’s modulating function approach was designed exactly for solving problems of this kind. It also solves another problem namely that of eliminating the unknown initial conditions for every interval estimate. The modulating function method, which has been introduced in the first part of this series (Preisig and Rippin, 1993a), also chooses a window of data; but since the initial conditions are eliminated, the dimension of the identification problem is reduced by the number of initial conditions, which is identical to the number of states. The spline-type modulating function, Male&sky’s extension of Shinbrot’s method, proved to be a very powerful method for solving this kind of problems. Applying the modulating function method to the model (see the Appendix), the following matrix equation is obtained which is discrete in time because. of the discrete sliding of the modulating functions. ARk)#k)
The vector and matrices
=
Wkll.
in this equation
P=(~)~=V$c, h,, h,, &I,
032) are
CD11
The parameters are obtained as a function of time from equation (E2) by solving for the parameter vector p for each set of data. Consequently, the parameters are obtained as a discrete signal from the identification stage with the same discrete time index as the modulated signals.
33
Theory and application of modulating function method-III 100 90
I-
jacket(average) _________ contents -
0
0.2
0.4
0.6
0.8
1
1.2
Time [s]
room
1.4
1.6
1.8
2! x104
Fig. 3. Stepwise heating and cooling.
4.3. Experiment The experiment was a combination of high- and low-frequency excitation covering a range of reasonable operating conditions. Several experiments were done. The most informative ones excited the jacket with a stair input increasing the temperature in the jacket stepwise by 10°C starting at a room temperature of about 20°C up to about 8O“C. Figure 3 shows the measured temperatures after correcting for the offsets in the measurements. The offsets were determined in a separate experiment and verified before the 6rst temperature step in this experiment. 4.4. Model rejnement The filtering properites were chosen to match the dominating time constant for the dynamic process of interest. Since the heat transfer through the wall dominates the dynamics of the overall heat distribution the major time constant of this subprocess was estimated and using as the characteristic time. A value of about 50 s was found to be feasible. The
tThe order of the modulating function v is usually chosen one bigger than the order of the model hecause the vth derivative of the modulating function, the stem function, is a set of Dirac delta functions. The stem functions, thus, use only the signal values that coincide with the Dkac impulses; all other information that lies in between is not utilized, which is the resson the stem function is usually not used (see Part I, !Iection 5.3).
choice is not critical. Variations of + 10 s do not affect the results to speak of. The order of the modulating function chosen is 2, which is one bigger than the highest order in the model equations.? Again, the choice of order does not affect the results visibly. The chioce of how much the modulating functions overlap or in other words the choice of the shift time AT has been discussed in Preisig and Rippin (1993a). A value of T/AT E [5. . . 101is a good rule of thumb. It should be noted that the shifting of the modulating function does not imply discrete processing of the measured data but only means that the output of the modulating filtering operation is discrete with a sampling time of AT. The results, shown in Fig. 4, have some interesting characteristics: (i) a trend between the HTP and the operating conditions is observed. With increasing temperature the HTP increase and vice versa; (ii) large, biased errors can be observed at the beginning of each step; and (iii) in the second phase, the cooling phase of the experiment, large errors are observed in the HTP. 4.4.1. Large errors in the cooling phase. The large errors in the cooling phase are due to a decreasing signal/noise ratio. As the process of heating reverts to cooling, the temperature profiles of the different elements cross each other. Since the driving force for the heat transfer is the temperature difference, no information remains in the temperature difference signal and the process cannot be identified. In other words, the system is not persistently excited (Astrom
H. A.
34
h@lslG and
D. W. T.
RIPP~N
1
-
jacket-contents ....... contents-gas phase ............. ize:-hd ..........
0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 C
D.2
a 1.4
0.6
0.8
1.2
1.4
1.6
1.8
Time Is]
2 x104
Fig. 4. The heat transfer parameters (HTP) as a function of time.
and Bohlin, 1966) and since the level of persistent excitation can only be estimated the interval during which the subprocesses are not identifiable can also only be estimated. The length of this interval depends on the noise level and the dynamics of the decrease
0.9
-
heating cooling
..._....
mdel
. ........ ...
and increase of the temperature difference. The large errors in the parameter estimates that are observed during the later part of the cooling phase are therefore not a weakness of the identification procedure but a consequence of the experimental conditions.
0.8 0.7
i
0.6 11.
D TetnpemmeConW Fig. 5. HTF’jacket--contents
80
[Cl
correlated with the temperature in the contents; empirical linear heat transfer parameter model.
Theory and application of modulatiug function method--III
35 I
Temperamre Gas Phase [Cl Fig. 6. HTF’ contents--gas
phase correlated with the temperature in the gas phase.; empirical exponential heat transfer parameter model.
4.4.2. Nonlinearities. The model parameters were identilkd using small sets of data. The data window, given by the length of the modulating function, is discretely shifted over the signals generating the discrete modulated signals. From these modulated
signals the parameters are calculated allowing them to slowly change with time and operating conditions. The changes, or trends, in the parameters are then used to extend the first simple model. The heat transfer parameters are graphically correlated with
-1
-2
-3
-4
-5 .... ... . ..,
-6
heating
mling ._______mdel
-7 20
30
40
50
60
70
Temperamre Gas Phase [Cl Fig. 7. HTP gas phase-lid
correlated with the temperature in the gas phase; empirical exponential heat transfer parameter model.
36
H.
A.
PREISIG
and D. W. T.
RIPPIN
0
I
. .. . -1 -
3
_________
3.1
3.2
hesting cooling model
3.3
3.4
3.5
3.6
3.7
3.8
3.9
ln( Temperature Difference (Lid-Room) [K] ) Fig. 8. HlT lid-room
correlated with the temperature difference between the. lid and the room; empirical power law heat transfer parameter model.
the operating conditions neglecting the biases and the intervals where the processes are not identifiable during the cooling phase. Figure 5 shows the HTP for the heat exchange between the jacket and the contents correlated with the temperature of the contents. The straight line represents the empirical function chosen for describing the correlation between the two quantities. Figures 6 and 7 show correlations between the HTP contents-gas phase and gas phase-lid, respectively. In both cases, an exponential relation seems to describe the behaviour of the HPT’s well. The actual heat transfer process is a mixture of heat conduction and convective evaporation and condensation. The conduction probably dominates at relatively low temperatures. As the temperature increases, the heat transfer due to evaporation and condensation increases until it completely dominates at boiling temperature. The last plot, Fig. 8, shows the correlation of the HTP for the heat exchange between the lid surface and the room. The empirical power law of natural convection was assumed to apply with a temperature difference raised to the poweri of I .25. As can be seen from the plot, this assumption fits the experimental data well. $This co&Cent includes the temperature dependency of the heat transfer constant.
4.4.3. Dynamic errors. The biased estimates obtained at the beginning of each step indicate that the model is not capable of describing the fast parts of the experiment accurately. Several problems can be identified. Firstly, the model describes the capacities in the area of the jacket with a single lump. Secondly, the model assumes uniform conditions in the jacket. Thirdly, heat transfer was assumed to occur between adjacent lumps only. Fourthly, the measurement devices are assumed to have no time lag. Using tbe nonlinear extensions identified above and comparing the temperature of the contents measured with the temperature reconstructed by the model, the problems with the dynamics become even more obvious. Also, even though special attention was paid to calibration of the temperature measurement devices, some calibration errors still remained as the difference does not approach zero at the end of the experiment. The calibration problem could not be corrected given the experimental set up (Fig. 9). The experiment presented in this paper was the best one obtained from this plant. The dynamic effects, however, were studied in separate experiments. The first one focused on the slow temperature measurements, mainly the sensor in the contents which, for mechanical reasons, is built into a solid support with a correspondingly high thermal capacity. Its time lag was estimated to fall within the range 20-60 s, dependent on the flow conditions.
37
Theory and application of modulating function method-III 1.5
1
0.5 ,
-Ieumulatlve error
-0.5
-1
0
200
400
600
800
1000
1200
1400
1600
1800
:
Cl
Fig. 9. Difference between the measured and the predicted temperature in the contents without dynamic corrections. The second experiment involved a small experimental setup, which consists of two tanks cut into a block Styrofoam separated by a wall of the same quality as the one in the reactor. Dynamic experiments were performed changing the temperature by
injecting hot water into one of the tanks which was half filled with water at room temperature. These experiments, discussed in more detail in Preisig (1984) and Koster (198 l), showed that a second-order model for the wall results in a dramatic improvement of the
time lag -0.5 -
6
c1
-1
0
-___-___...
0.2
0.4
0.6
1
0.8
Time [s]
1.2
1.4
ZS 40 s
1.6
1.8
2 x104
Fig. 10. Difference between the measured and the predicted temperature in the contents, with dynamic
wrreetions.
H.
38
A.
hEISIG
and D. W. T.
RPPIN
Table 1 COMtitiOtM
Steadv-state wints 43.5 37 32 22
53.5 44 38 22
63.5 52 49 22
73.5 63 61 22
83.5 18 77 22
Measured heat losses Estimated heat losses
0.15 0.17
0.22 0.27
0.53 0.40
0.59 0.59
0.89 0.87
dynamic description. This approach has therefore been adopted in this study. The fluid-film resistances on the two sides of the wall are assumed to be of equal magnitude, knowing the resistance introduced by the wall because of known physical properties and geometry. Figure 10 shows the comparison between the measured temperature and the model responses with different time lags for the temperature sensor in the contents. Including the sensor dynamics improves the results significantly. The error is reduced by a factor of two and remains over the whole experiment within the limits of kO.6 K. 4.4.4. Steady state errors. Errors in the steady-state description would, to a certain extent, accumulate in an observer scheme. Therefore, the accuracy of the model in describing the steady-state behaviour of the plant, mainly the heat losses, is important. For the description of the heat losses through the lid, a number of ad hoc assumptions have been made and must be checked by comparing the predicted steady-state heat losses and the measured heat losses. The steady-state temperatures in the gas phase and on the surface of the lid can be calculated from the model by zeroing the accumulation terms on the left-hand side and substituting the models for the HTP parameters resulting in two nonlinear equations in the gas and lid temperature. The two equations are:
=fl exp&y,l(~,
-n)
- ~I,IY, -Y~I~~~~(Y~ -Y,) = 0, (E3)
where ai,jv&j
::
Units
Temperature contents Estimated temperature gas phase Estimated temperature lid Room tmmxrature
the parameters in the exponential relations describing the heat transfer between lump i, j.
These equations are readily solved for the steady-state temperatures using a Newton-Raphson iteration scheme. Table 1 shows the comparison of predicted steady-state heat losses and values measured in an experiment that had been performed under very similar operating conditions about a year earlier. The accuracy with which the model describes the steady-state heat losses is extremely good, only the value at y. = 63.5”C deviates significantly.
“C DC “C “C
Lastly, the steady-state analysis can be used to estimate the errors between the dynamic description and the steady-state description. Figure 11 shows a comparison of the predicted steady-state temperature differences and the measured values. The differences between the measured and the steady-state predictions are quite small during the heating phase. However, the differences are much bigger during the cooling phase. These graphs show the magnitude of the dynamic errors that have to be expected. This more sluggish behaviour of the process to cooling is confirmed by practitioners operating industrial plants. The quality of the identified model can also been judged by analyzing the accuracy of the simulated data using the model outside the range of operating conditions used for the identification of the model. The temperature differences between the contents, the gas phase and the lid decrease rapidly with increasing system temperature, as one would expect. 5.
SIGNIFICANCE
AND CONCLUSIONS
The pilot plant for which a dynamic description of the heat dissipation was sought had all the attributes that are typical for an industrial plant: the heat capacity of the construction elements are not negligible and of a distributed nature, the measurements are not very accurate and the heat losses are significant depending on the operating conditions. These are all reasons for which it is literally impossible to formulate and identify a very detailed model of the process. In reviewing the results gained in this investigation the authors arrived at the opinion that the approach of successively refining a simple model offers an attrao tive solution to the problem of identifying a dynamic model for a badly defined system. The modulating function method-in particular Maletinsky’s splinetype modulating functions--proved to be a powerful tool for evaluating the parameters as the process proceeds. Correlating the parameter variations induced by modelling errors with the changing state variables results in nonlinear extensions of the original simple model. At the present time, graphical analysis appears to be the most appropriate way of determining these extensions since it enables the analyst to qualitatively weigh the different sections of the experiment.
Theory and application of modulating function method-III 50
39
I
40-
contents-gas phase
heating cooling
20
30
40
50
60
70
80
90
100
Temperature Contents [Cl Fig. Il. Measured
vs calculated
Considerable improvement in model accuracy can be achieved by simple dynamic extensions that satisfy steady-state conditions. The steady-state heat losses predicted by the extended model compare excellently with experimental data obtained from the plant. REFRmNCES
Astriirn K. J. and T. Bohlin, Numerical identification of linear dynamic systems from normal operating records. IFAC Symp. Theory of Self-Adaptive Control Systems, Teddington. England in Theory of Self-Aakptive Control Systonr (Edited by P. H. Hammond). Plenum Press, New York (1966). De Valli&e P. R. L., State and parameter estimation in batch reactors for the purpose of inferring on-line the rate of heat production. Ph.D. Thesis, Swiss Federal Institute of Technology, Diss ETH Nr 8847, Zurich (1989). Kalman R. E. and R. S. Bucy, New results in filtering and prediction theory. J. Basic Engng Man, (1961). Koster U., Identi8kation einfacher Modelle filr einen verteilten W&medurchgang; Diploma Thesis No. 3900, Swiss Federal Institute of Technology (ETH), Tech. Chem. Lab. (1981). Luenberger D. G., An introduction to observers. IEEE Trans AC Control AC-16, (1971). Preisig H. A., On the identification of structurally simple dynamical models for the energy distribution in stirredtank reactor equipment. Ph.D. Thesis, Swiss Federal Institute of Technology, Diss ETH Nr 7616, Zurich (1984). Preisig H. A., The use of differential information for batch reactor control. ACC 88, Atlanta (1988). Preisig H. A., On-line observation of the composition in
steady-state
temperature
differences.
non-isothermal, batch reactors with nonlinear reactions. ACC 89, Pittsburgh (1989a). Preisig H. A., Effect of modelling errors on the observation of concentrations in batch reactors. AIChE Annual Meeting, 1989, Paper 167Bq, San Francisco (1989b). Preisig H. A., Application of observer using differential information to bench-scale reactor. ACC 90, San Diego (1990). Preisig H. A. and D. W. T. Rippin, Theory and application of the modulating function method-I. Review and theory of the method and theory of the spline-type modulating functions. Computers them. Engng 17, l-16 (1993a). Preisig H. A. and D. W. T. Rippin, Theory and application of the modulating function method-II. Algebraic representation of Maletinsky’s spline-type modulating functions. Computers &em. Engng 17, 17-28 (1993b). Wallman P. H. and A. S. Foss, Experiences with dynamic estimators for fixed-bed reactors. Ind. Engng Chem. Funaiun. 20, (1981). Weber R. and C. Brosilow, The use of secondary measurements to improve control. AIChE JI (1972).
APPENDIX Modulation of the model equations (El) (multiplication of the set of equations with the modulating integration by parts) results in: qs=
-h,(jl,-k)+q(jW,.
function and
p),
e,~==~(q-~~)-h,(~~-_3r), c‘~~=Eh,(~=-~~)-4(4-A), c,? =h~(~*-a)-ht(a-~~). (E4) Rewriting in matrix form and substituting the definitions (Dl), (D2) (D3) and (D4) equation (E2) is obtained.