274
Ind. Eng. Chem. Res. 1992,31, 274-281
On-Line, Closed-Loop Identification of Multivariable Systems Didgenes L. Melo and John C. Friedly* Department of Chemical Engineering, University of Rochester, Rochester, New York 14627
A new method to determine the open-loop transfer functions in an interactive N X N linear multivariable system based on a simple experimental test, for example, a step change in the controller set point carried out during closed-loop proportional integral derivative (PID) control operation, is presented. Simulation results for 2 X 2 , 3 X 3, and 4 X 4 systems over a frequency range adequate for control purposes show good agreement between the transfer functions obtained from the experimental test and the transfer functions used in the simulations, even for very strongly interactive cases. The method also proved to be very robust with regard to measurement noise and load disturbances that may enter the system during the closed-loop test. A nonparametric model obtained by this technique can then be used directly in a PID control algorithm or an internal model control configuration or as the basis of a parametric model more useful for dynamic matrix control for example. Introduction In the past 10 years there has been a tremendous upsurge in process control interest both in industry and in academia. Much of that has been focused on new developments in model predictive control. Viable alternatives to the classical PID control such as dynamic matrix control and model algorithmic control have been developed for practical industrial problems, permitting the control engineer to address in a reasonable way some of the realistic complications that must be treated in any practical control system. A wide variety of new model predictive approaches have been suggested and a unifying theme has been provided by the work on internal model control. The use of any of the model predictive control techniques involves at least three important aspects. The model must be identified, the control algorithm determined, and the robustness of the system maintained. Inherent in the use of any model predictive technique is the model itself. The model must be identified in either an on-line manner as is done in adaptive control applications or in model algorithmic control or in an off-line manner as is done in dynamic matrix control for example. The models used can be parametric or nonparametric, depending on whether the structure of the model is chosen in advance. Adaptivq control algorithms generally adapt parameters in a prechosen model structure. On the other hand, the frequency response characteristics used in design of robust internal model control system are nonparametric. Process models are never perfect and because of unmodeled nonlinearities or drifts with time need modification from time to time. This can be done continuously in a recursive manner as in adaptive control techniques or periodically in step or impulse testing. In either case there is great advantage to being able to conduct the test without disturbing the process excessively. This necessarily involves a trade-off between providing a sufficiently disturbing signal with a high enough signal-to-noiseratio and maintaining continuous control of the process. The class of identification techniques performed while the process remaina in closed-loop control has great advantage over open-loop tests because the process can be run continuously throughout the test, even though it may be disturbed somewhat from the nominal set point for example. Literature Review A fundamental stage in the design of a model predictive controller is the formulation or identification of a process model. Control techniques such as model algorithmic
control, Mehra et al. (1982) and Richalet et al. (1978); inferential control, Joseph and Brosilow (1978) and Brosilow and Tong (1978);dynamic matrix control, Cutlp and Ramaker (1979); and internal model control, Garcia and Morari (1982), explicitly require the use of a process model. There are two ways to obtain a model for a process for control applications. First, it may be possible to obtain a model by detailed analysis, writing the differential or difference equations that arise from the application of the mass, energy, and momentum conservation equations; or second, it is possible to determine a model from observations of the input and output of the plant. In this work we will refer only to the second approach. Extensive work has been done in the past 20 years on the subject of system identification and several boob have been written, for example, Lee (19641, Davies (1970), Mendel (19731, Eykhoff (19741, Graupe (19751, Mehra and Lainiotis (19761, Goodwin and Payne (19771, Hsia (1977), Sinha and Kuszta (19831, Ljung (19871, and Unbehauen and Rao (1987). Also several,survey or review papers have been written on the field: Astr6m and Eykhoff (1971), Gustavsson et al. (19771, Kubnrsly (19771, Billings (1980), Godfrey (1980), Ljung and Glover (19811, Wellstsad (1981), Young (19811, Prett et al. (1987), and Unbehauen and Rao (1990). A recent paper by Rivera et al. (1990) discusses the subject of control-relevant identification, fqFusing specifically on system identification for control design purposes. Identification methods can be divided into nonparametric and parametric types. Parametric identification methods primarily are based on prescribed model forms and include parameter estimation methods for exampie. Nonparametric identification methods primarily yield forms such as frequency responses. Some parametric methods lead directly to parametric models, but others result first in a nonparametric model and then in a parametric model. Most of the literature previously mentioned has focused on methods that result directly in parametric models. There are several methods for identification of nonparametric models [Godfrey (1980); Wellstead (198l)l. Different approaches have been applied to estimate transfer functions from frequency responses [Amman (1964); Jong and Shanmugam (1977); Shieh and Cohen (1978); Lin and Wu (1982); Braun and Ram (1987); Sidman et al. (1990)l. Linear least-squares methods for transfer function synthesis from frequency response data has been discussed by Levy (1959), Sanathanan and Koerner (1963),Payne (1970), Lawrence and Rogers (19791, Stahl (1984), and Whitfield (1986).
0~-5~5/92/2631-0274$03.00/0 0 1992 American Chemical Society
Ind. Eng. Chem. Res., Vol. 31, No. 1,1992 271
/Ails O
I X(s) =
(C(l)(S) c'2'(s)
*..
... 0 .. .
\
C(n)(S))
Now we can write (3) as Figure 1. Block diagram of a multivariablefeedback control system.
X = [(I + GG,)-'GG,]X,
Closed-loop identification methods are discussed by Gustavsson et al. (1977), and in general are given far less attention than open-loop methods. Perhaps this is becaw it is easily shown that in the closed loop sometimes it can be impossible to identify some of the parameters [Gustavsson et al. (1977); Ljung (1987)l. In spite of that limitation, nonparametric models can be obtained. However, there is no sign at all of any experimentally simple, nonparametric method that works in a closed-loop mode, works for a given PID controller setting, and is able to identify the open-loop transfer functions of a multivariable process, as the method presented in this work does.
We can solve for G from (4): G = XXL1(I - XXL')-'G;'
Identification Method In this section we will present the new method developed to identify the open-loop transfer functions in an interactive N X N multivariable system. The method is based on a simple experimental test, for example, a step change in the controller set point, and it is carried out during closed-loop PID control operation. Several examples using the method will be discussed. Consider the following N X N multivariable system operating under PID control shown in Figure 1. Aasuming the disturbances Ut) equal to zero, we can write in the Laplace domain c=Gm (1) m = G,(r - c)
(2)
where c is the vedor of the output variables, r is the vector of set-point variables, m is the vector of the controlled Variables, G is the matrix of process transfer functions, and G, is the matrix of the feedback controller functions: c =
(!l);
r =
(p1 811
G =
812
**.
gn2
* * a
m =
gnn
G, =
(:'> mn
(: $" ::: ) Bell
gln
. . ::'. g. b ) ;
Bnl
(:I);
rn
Cn
0
e..
0
0
* e .
gcnn
Combining (1)and (2) and solving for c: c = [(I + GG,)-'GG,]r
(4)
(5) Equation 5 then permits the evaluation of the multivariable process transfer functions in terms of the known set of controller transfer functions G,, the input set point changes X,, and their corresponding measured output responses X. Any suitable numerical technique for obtaining Laplace or Fourier transforms may be used to obtain the frequency responses from the measured time responses. We have chosen the following way. In the Laplace domain, every element of (3) is of the form
Recalling that the Laplace transform of the derivative of a function is just sl:V(t)], (6) can also be written
J -dc:t)
-c,li)(s) rpqs)
- --
e-st dt
SC?)(S)
-
sr?)(s)
(7) J-&y;t)
e-8t dt
If the set point change is a perfect step of magnitude A,, the denominator of (7) can be evaluated and the frequency response of c?(s) obtained
To evaluate (8),we used a centered finite-divided-difference formula retaining the second derivative term from the Taylor series expansion to calculate numerically the derivative and the fast Fourier transform algorithm, Brighman (1974))to carry out the integration. The alternative approach of integrating (8) by parts to avoid numerically differentiating the data yields [Schechter and Wissler (1959); Nyquist et al. (1963)]
(3)
For any given PID controller G,, if we introduce a step change of magnitude AI into the set point for rl and we keep track of the system response with time, then rl(s) = Al/s; c(t) = measured; G, = known We repeat this procedure for r2(s) and so on up to rJs). In each of the set of n experiments we make a different set point change, designated rG)(s),and obtain a corresponding vector cO(t). We then numerically evaluate the Laplace transform of the response to obtain c%). In this way then, we can form matrices that represent these experiments. Let X,(s) be the matrix for the set point changes and X(s) be the matrix for the corresponding output values
However, it has been shown by Rajakumar and Krishnaswamy (1975) that the numerical differentiation procedure as applied to exponential type of step responses produces better values at high frequencies than the ones obtained using (9). Our simulation study c o n f i i e d this result.
Examples Let us illustrate the use of the method on three simulation examples. The first example is the two input-two
276 Ind. Eng. Chem. Res., Vol. 31, No. 1,1992 1.4
0.2
I
1I..-"
OOY 0
0.02 25
75
50
0.01
100
0.1
Time 1
0.8,
".-
1
Frequency w
b
,
-450
25
is
5.0
160
Time Figure 2. Simulation of a binary distillation column, using the Wood and Berry (1973)model. Set point change of magnitude 1.0 in rl(t). The solid line shows the response using proportional integral controllers with original Wood and Berry (1973)settings, and the dotted line shows response using proportional controllers.
0.01
0.1
1
10
Frequency w Figure 3. Simulation of a binary distillation column, using the Wood and Berry (1973)model. Bode plot for gll. The solid line represents the original transfer function, and the points are the identified frequency response calculated by the new method. Squares represent results using PI controllers and crosses those using P controllers.
output distillation column given by Wood and Berry (1973). The process open-loop transfer functions are
In Figure 2, there is a representation of the closed-loop response of cl(t)and c2(t) for a step change of magnitude 1.0 in rl(t). The solid line represents PI controllers with the settings originally used by Wood and Berry (1973): 0.2(1 + G,(s) =
1 1 4.44s
0.014 0.01
0.1
Frequency o
I
I
10
1
10
0
-0.04(1 +
2.67s
The dotted line represents proportional controllers with gains selected to give reasonably damped closed-loop responses:
F
-180-
-380-450
0.01
With these proportional controllers the responses of both output variables for a step change in r2(t)are both delayed monotones more like cl(t) in Figure 2. For the PI controllers the responses for a step change in r 2 ( t )are both oscillatory. For all the simulations with this example, 2048 sample points were used at a sampling rate of 0.1 time unit. Applying (8) and calculating G(io) from (8,we can construct the Bode plot for each of the four transfer functions. Two of these, for the direct response gll(iw)and for one of the interaction terms g21(io),are shown in Figures 3 and 4. The solid line repreaents the original transfer functions given by (111, and the points are calculated by using (5). The squares were obtained using the PI controllers and the crosses were obtained using the P controllers. Clearly there is excellent agreement between both predicted and original transfer functions over a frequency range quite acceptable for most control applications. At
0.1
Frequency w Figure 4. Simulation of a binary distillation column, uaing the Wood and Berry (1973)model. Bode plot for gZ1. The solid line represents the original transfer function, and the points are the identified frequency response calculated by the new method. Squares represent results using PI controllers and cro88B8 those using P controllers.
higher frequencies than shown in Figures 3 and 4 a clear breakdown of accuracy is evident. The other two transfer functions are obtained with accuracy comparable to Figures 3 and 4 over the same frequency range. In constructing the Bode plota, and in order to avoid congestion of points at high frequencies, only a fraction of the points were actually plotted at frequencies higher than 0.2. In order to study how the method responds to an unmeasured or unanticipated disturbance, a disturbance load was introduced into the system at the beginning of the experimental test. For each set point change in one loop,
Ind. Eng. Chem. Res., Vol. 31, No. 1,1992 277
0.024 0.01
0.1
1
.I
1
0.02 0.01
10
Frequency w
0.1
Requency
D
e3
-180
t
-270
e
1
10
1
10
GJ
6
-360 -450
Requency w Figure 5. Simulation of a binary distillation column, using the Wood and Berry (1973)model. Bode plot for gll. Effect of unmeasured disturbances. Disturbance is a step change of magnitude 0.2. Squares represent results using PI controllers and crosses those using P controllers.
0.1
0.01
Frequency o Figure 7. Simulation of a binary distillation column, using the Wood and Berry (1973)model. Bode plot for gll. Effect of unmeasured disturbances. Disturbance is a ramp of magnitude time X 0.05. Squares represent results using PI controllers and crosses those using P controllers. X
1
0.014 0.01
0.1
1
. . I 10
Requency w
“ 1
0.014 0.01
0.1
0 -90
$
-180
p)
-no
e
z 2
.I
10
Requency w
0
8
1
0
-360 -450
0.01
0.1
1
10
5
-460 0.01
0.1
10
Frequency w Figure 6. Simulation of a binary distillation column, using the Wood and Berry (1973)model. Bode plot for gzl. Effect of unmeasured diaturbances. Disturbance is a step change of magnitude 0.2. Squarea represent results using PI controllers and crosses those using P controllers.
Requency o Figure 8. Simulation of a binary distillation column, using the Wood and Berry (1973)model. Bode plot for gZ1. Effect of unmeasured disturbances. Disturbance is a ramp of magnitude time X 0.05. Squares represent results using PI controllers and crosses those using P controllers.
a single disturbance load change was made in the same loop. As before, the set point change was a step of magnitude 1.0. In the first case, the disturbance was a step change of magnitude 0.2 (20% of the set point change) introduced at the output of the process, U t ) in Figure 1. In Figures 5 and 6 there is the Bode representation for the transfer function gll(iw) and g,,(iw) obtained under these conditions. In the figures the solid line represents the open-loop transfer function given by (11)and the points are the frequency response obtained by using (5) for both P and PI controller settings. In the second case, the disturbance was a ramp of magnitude time X 0.05, and the
corresponding results are presented in Figures 7 and 8 for the transfer function g,,(iw) and gzl(iw) for both P and PI controller settings. All four figures show good agreement for the moderate to high frequency range shown. Unmeasured step and ramp disturbances cause more deviation from the exact response at the low frequency. In these figures, the identified responses using the PI controllers are somewhat worse at low frequency than those using the P controllers. This is believed to be due to the oscillatory nature of the responses shown in Figure 2 when the PI controller settings were used. Other simulations have shown that when the responses using either PI or P con-
278 Ind. Eng. Chem. Res., Vol. 31, No. 1, 1992
h
0.8 0.4
0
25
50
75
100
75
Id0
0.024 0.01
0.1
Time
Fkequency w
1
1
10
1
10
na
i
-450
25
50
Time Figure 9. Simulation of a binary distillation column, using the Wood and Berry (1973)model. Set point change of magnitude 1.0 in rl(t). Effect of noise contaminated data on the system response using PI controllers.
Frequency w
Figure 10. Simulation of a binary distillation column, using the Wood and Berry (1973)model. Bode plot for g,, obtained using the noisy data with no filtering. PI controller settings. '01-
troller settings are less oscillatory, the agreement a t low frequency is improved. On Figure 7 for the ramp disturbance, the g,, transfer function also shows the beginning of a breakdown in accuracy at the higher frequency as well. However the g,, transfer function on Figure 8 does not yet show the same effect in this frequency range. It should be kept in mind that the unmeasured ramp disturbance has grown to a magnitude of about 10 times the step change in set point by the end of the time span of 200 time units used for identification. In spite of that huge unmeasured disturbance the identification method works quite well. In all four figures the frequency response is identified sufficiently well in the frequency range of importance, covering the response up to the point that the phase angle is well below - 1 8 0 O . In general, these results, as well as those of other simulations, show that the method copes in a very reasonable fashion with the unmeasured disturbance load. Comparable resulta were obtained when the disturbance occurs for both loops. If the disturbance is known or measured, it should be included in the calculations; i.e., one should include one more term into (1). Another test of the goodness of the new method is the introduction of noise into the measured variable. We introduced at each output variable of the process a random noise from a normally distributed deviate with zero mean and variance of 0.1. In Figure 9 there is a representation of the noisy output for the PI controller settings, where the solid line represents the noisy data. The signal-to-noise ratio is of the order of 3 or 4 to 1in this case. In Figures 10 and 11we can see the agreement achieved between the calculated values using the noisy data directly with no filtering and (5), represented by the points in the plots, and the theoretical transfer functions using (ll), the solid line in the plots. As the frequency increases, the effect of the noise on the calculated values is more evident. Still, there is adequate agreement for all frequencies up to at least 0.5, giving an accurate transfer function up to a phase shift of nearly - W O O , sufficient for many control applications. If a first-order filter with unity steady-state gain is used, the good agreement can be extended to higher frequencies, making the method even more robust. For
0.1
0.01
Y
I
.4
O.li
0.014 0.01
2
-180-
0)
-270-
:
e
0.1
Fkequency w
I
1
10
\ -
I10
-380-450
4
0.01
0.1
I
Fkequency w Figure 11. Simulation of a binary distillation column, using the Wood and Berry (1973)model. Bode plot for g,, obtained using the noisy data with no filtering. PI controller settings.
example, using a filter time constant equal to 10 increases the frequency range of good agreement by a factor of 4. Similar reaulta are obtained for the case with P controllers. If the magnitude of the noise introduced into the system is larger, the effect of the noisy data on the calculated transfer function values appears at lower frequency values than those in Figures 10 and 11. For noisier processes it is necessary to filter the data before using this identification method. The problem of noise domination at high frequencies is well known. [See for example Jansson (19841.1 However, it is believed that noise is less of a problem in these examples than in problems involving numerical deconvolution discussed by Jansson because of less need for accuracy at higher frequencies. The second simulation example studied is a 3 X 3 system attributed by Luyben (1986) to Ogunnaike and Ray (1979).
Ind. Eng. Chem. Res., Vol. 31, No. 1, 1992 279
0'4 0.3
0.0
'
!
0
10
20
I
30
40
50
30
40
50
30
40
50
Time
!
0.0
I
0
20
10
1
Time
!
-15.0
0
10
20
4
Time Figure 12. Simulation of a 3 X 3 distillation column, using the Ogunnaike and Ray (1979) model. Response to set point change of magnitude 1.0 in rl(t) using P controller settings.
Figure 14. Simulation of a 4 X 4 distillation column, using the Doukas and Luyben (1978) model. Bode plot for gw The solid line represents the original transfer function, and the points are the identified frequency response calculated by the new method using P controller settings.
in r2(t) and r3(t)are similar in form. For simplicity we used a diagonal proportional controller with gains:
0.002
-450
0.01
0.1
0.01
0.1
hquency w
I
1
10
1
10
Frequency w Figure 13. Simulation of a 3 X 3 distillation column, using the Ogunnaike and Ray (1979) model. Bode plot for gw The solid line represents the original transfer function, and the pointa are the identified frequency response calculated by the new method using P controller settings.
The process open-loop transfer functions as reported by Luyben (1986) are
G(d=(
-0.61e4.S. -0.0049e-''08 9.06s + 1 8.64s + 1 ~ l l e " . ~ - 2 . 3 ~ ~ a.ole-lA 3.26s + 1 5.0s + 1 7.09s + 1 44.68e4% 46.2e4'& 0.87(11.61s+ l)e-l'o. (3.8% + lX18.8s + 1) 8.16s + 1 10.9s + 1
- - -
gel1 = 1.0 gc22 = -0.1 gc33 = 1.0 Figure 13 shows the Bode plot for the open-loop process a first order lead, second order transfer function gS3(iw), lag transfer function. The solid line represents (12), and the points are calculated by using (5). Again, the agreement between these two is excellent, suggesting that the identification method is readily applicable for higher order systems. All nine transfer functions of this process are obtained with comparable accuracy. The last example is the simulation of a ternary distillation column using the four input-four output system given by Doukas and Luyben (1978). The process openloop transfer functions are
as)= -9.8 1le-l.'" 11.368 + 1 6.984eaa 14.29s + 1 2.38e4.a (1.43s + 1Y -11.672e-"t 12.19s + 1
\
\
1 (12)
In Figure 12 there is a representation of the closed-loop response of cl(t), c&), and c3(t) for a step change of magnitude 1.0 in rl(t). The responses for set point changes
0.37&-'.'~ 22.228 + 1 -1.986e4'.71' 66.67s + 1 0.0204e-9"" (7.148 + 1p - 0 . 1 7 6 ~ (6.9s + 1?
-2.368e41.3" 33.3.9 + 1 0.422e"'72. (250.0s+ 1,s 0.61%-'" 1.0s + 1 16.64e-''0. ~ ~ 1.08 + 1
-11.3e3.m (21.748 + 1Y 400.0s+ 1
(2.38s + 1Y 4.48e-9.52. 1l.lls + 1
(13)
Again, for simplicity we used only proportional controllers with gains:
280 Ind. Eng. Chem. Res., Vol. 31, No. 1, 1992
In Figure 14, there is the Bode plot for the open-loop process transfer function g,(iw). The solid line represents (13), and the points are calculated by using (5). Again, the agreement between these two is excellent. All 16 transfer functions of this process are obtained with comparable accuracy. It is important to realize the wide range in parameter values in this system, with time constants ranging from 1 to 400 and delay times ranging from 0.4 to 60. All transfer functions were identified by this method. These three examples give results representative of those obtained for a variety of simulated models and conditions. The identification technique is capable of providing a nonparametric frequency response for all transfer functions of a multivariable process over a frequency range adequate for most control applications. The method gives results which clearly break down at higher frequencies, typically like that shown in Figures 10 and 11. Therefore it should be possible to determine from the results themselves the frequency above which the model will not be useful. The method is fairly robust to unmeasured disturbances and to noisy data. The frequency range over which a useful model is obtained is decreased as the signal-to-noise ratio decreases, but filtering the data in advance tends to counter the effect of noise. Although noise may prevent this simple approach from being used for the higher frequency range and thus to obtain accurate parametric models based on a larger frequency range, it has been found to give frequency response models sufficiently accurate for control purposes over a wide range of problems and conditions.
Summary and Conclusions A relatively straightforward method for the on-line identification of all open-loop transfer functions of a multivariable process is presented. The method is applied while the process is under closed-loop control, so that any upset from normal operations is minimized. The method provides a nonparametric, frequency response model that is sufficiently accurate for most control system designs. The model obtained can be used as it is or as the basis for a parametric model to be incorporated in a model predictive control environment for example. Numerical simulations, representative examples of which have been presented here, have shown that the method is sufficiently robust in the face of unmeasured disturbances and noise. Low-frequency results tend to be better if the controllers are tuned to give an overdamped response. The resulting models are quite accurate for low frequencies, even using raw noisy data, and it is clear from the results at which frequency the method starts to fail. As to be expected, the range of accurate results decreases with decreasing signal-to-noise ratio. However, signal conditioning with simple digital filters can be used to broaden the frequency range for which the model can be obtained accurately. In all cases studied in the presence of noise it was possible to obtain a sufficiently accurate representation of the simulation models up to the frequency corresponding to a -180O phase lag. Literature Cited k 6 m , K. J.; Eykhoff, P. System Identification-A Survey. Automatica 1971, 7, 123-162. Ausman, J. S. Amplitude Frequency Response Analysis and Synthesis of Unfactored Transfer Functions. Trans. ASME. Ser. D. J. Basic Eng. 1964, 86, 32-36. Billings, S. A. Identification of nonlinear systems-a survey. IEE Proc., Part D 1980, 127, 272-285. Braun, S. G.; Ram, Y. M. Structural Parameter Identification in the Frequency Domain: The Use of Overdetermined Systems. Trans.
ASME. J . Dyn. Syst., Meas., Control 1987, 109, 120-123. Brighman, E. 0. The Fast Fourier Transform; Prentice-Hak Englewood Cliffs, 1974. Brosilow, C. B.; Tong, M. Inferential Control of Processes: Part 11. The Structure and Dynamics of Inferential Control Systems. AZChE J. 1978,24,492-500. Cutler, C. R.; Ramaker, B. L. Dynamic Matrix Control-A Computer Control Algorithm. Presented at the AIChE 86th National Meeting, Houston, TX, 1979. Davies, W. D. T. System Identification for Self-Adaptive Control; Wiley-Interscience: London, 1970. Doukas, N.; Luyben, W. L. Control of Sidestream Columns Separating Ternary Mixtures. Instrum. Technol. 1978, 25, 43-48. Eya-off, P. System Identification; Wiley: New York, 1974. Garcia, C. E.; Morari, M. Internal Model Control. 1. A Unifying Review and Some New Results. Ind. Eng. Chem. Process Des. Dev. 1982,21,308-323. Godfrey, K. R. Correlation methods. Automatica 1980,16,527-534. Goodwin, G. C.; Payne, R. L. Dynamic System Identification, Experiment Design and Data Analysis; Academic Press: New York, 1977.
Graupe, D. Identification of Systems; Kreiger: New York, 1975. Gustavsson, I.; Ljung, L.; SMerstr6m, T. Survey Paper: Identification of Process in Closed Loop-Identifiability and Accuracy Aspects. Automatica 1977, 13,59-75. Hsia, T. C. System Identification; Lexington Books: Lexington, MA, 1977.
Jansson, P. A. Deconuolution With Applications in Spectroscopy; Academic Press: Orlando, 1984. Jong, M. T.; Shanmugam, K. S. Determination of a transfer function from amplitude frequency response data, Int. J. Control 1977,25, 941-948.
Joseph, B.; Brosilow, C. B. Inferential Control of Processes: Part I. Steady State Analysis and Design. AZChE J. 1978,24,485-492. Kubrusly, C. S. Distributed parameter system identification-a survey. Znt. J. Control 1977,26,509-523. Lawrence, P. J.; Rogers, G. J. Sequential transfer function synthesis from measured data. Proc. ZEE 1979, 126, 104-106. Lee, R. C. K. Optimal Estimation, Identification and Control; MIT Press: Cambridge, 1964. Levy, E. C. Complex curve filtering. IRE Trans. Autom. Control 1959,4,37-43.
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Received for review December 13, 1990 Revised manuscript received June 17, 1991 Accepted August 15, 1991
Dynamic Simulation of a Supercritical Fluid Extraction Process Balshekar Ramchandran, James B. Riggs,* and Hubert
R. Heichelheim
Department of Chemical Engineering, Texas Tech University, Lubbock, Texas 79409-3121
A. Frank Seibert and James R.Fair Separations Research Program, The University of Texas a t Austin, Austin, Texas 78712
Supercritical fluid extraction (SFE) is a process in which a solute present in a homogeneous mixture is preferentially dissolved by a supercritical fluid (SCF) solvent. SCFs are gases or liquids above their critical points. Under these conditions SCFs exhibit enhanced solvating power, higher selectivities, and other desirable physicochemical properties which makes them particularly good solvents. A dynamic SFE process simulator has been developed by modeling various units of a typical SFE process, and realism of the simulated results is enhanced by accounting for instrument noise, random drifts, transport delays, and analyzer dead times. The process simulator has been compared with operating data from a pilobscale SFE process and yields good agreement. The dynamic process simulator provides a much needed insight into various process interactions and the nonlinearities that can exist in an integrated, multiunit process.
Introduction In recent years, the continued impact of energy conservation measures has resulted in a trend toward increased attention to economics, safe operations, and environmental consciousness. The above-mentioned factors have driven engineers and designers toward tighter integration of process plants catalyzed by the ever-increasing cost of capital. As Luyben (Tzouanas et al., 1990) puts it ’’...economic need to operate industrial processes as close as possible to optimum specifications with minimum energy consumption while safety and environmental constraints are not violated has produced processes that are more complicated...” Supercritical fluid extraction (SFE) has been hailed to have the potential to solve a number of challenging separationsproblems, and the past five years have seen a great deal of innovation in using SFE for treatment of foods, fractionation of pharmaceuticals, purification and impregnation of polymers, manipulation of polymer morphology and porosity, and extraction of hazardous wastes. There is a definite hesitancy on the part of industry to apply SCF technology because of the high capital costs involved, relatively little known process phenomena and thermodynamic behavior of SCFs, and a *Towhom all correspondence should be addressed. 0888-5885 192/263l-O28l$O3.W JO
lack of demonstrated control (Energy Conservation, 1989). The application of a SCF as a solvent is based on the experimental observation that many gases exhibit enhanced solvating power at conditions above the critical point. The solvating power of a SCF solvent can be directly related to the density of the solvent in the critical region and is very strongly dependent on the pressure and temperature. With adjustments of pressure and temperature, the solvating power of the SCFs can be fine-tuned to display a wide spectrum of solvent characteristics. Even though the literature on SFE is extensive, it is very sketchy because process applications have been developed as the understanding of the fundamental behavior of SCFs increased. Unfortunately, many of the best successes are proprietary, such as the applications in foods, flavors, and fragrances and the difficult separations of thermally labile pharmaceuticals; for the most part, information on these applications is not yet available in the open literature. More reviews on process application are available elsewhere (Williams, 1981; Paulaitis et al., 1982; Ely and Baker, 1983; Rizvi et al., 1986; Larson and King, 1986; Eckert et al., 1986, Johnston and Penninger, 1989; Cygnarowicz and Seider, 1991). One of the best is by McHugh and Krukonis (1986), who have put together a comprehensive overview on the SFE process principles and practice, discussing the process, the thermodynamics and phase behavior modeling, 0 1992 American Chemical Society
