Ind. Eng. Chem. Res. 1993,32, 228-230
228
RESEARCH NOTES Estimation of the Second-Order Parameters from the Process Transient by Simple Calculation A calculation method to estimate the parameters of the second-order-plus-dead-time model from the process transient is presented in this study. Parameters are estimated from four points of the step-response data using some correlation equations without any graphic technique or computer searching. This method is much easier to apply than the earlier methods. Introduction
f3(5)
To develop a mathematical model for a chemical process is often the first-step undertaken in the design of a controller. It has been recognized that most of the process dynamics may in general be simplified by the first- or the second-order-plus-dead-timemodel. Previous approaches involving the second-order parameter estimation, without computer searching, from stepresponse data have been attempted by Oldenbourg and Sartorius (1948), Meyer et al. (1967), Sten (1970), Sundaresan et al. (1978), Huang and Clementa (1982), etc. Other methods can also be found in recent process control textbooks, such as those by Seborg et al. (1989) and Coughanowr (1991). One of the drawbacks common to these available methods is that they are in reliance upon the graphical techniques. The aim of this paper is, therefore, to develop a reliable method for estimation of the second-order parameters from step-response data by simple calculation without using any graphical technique. This method is considered to be easily implemented in a pocket calculator. It can also be easily programmed in a digital computer to provide close initial guesses for nonlinear least-squares fitting.
= t3/T = 0.848967 + 0.071809(
+ 0.19753t2 - 0.021823t3 (3)
(SE = 0.497 X f6(5)
r = l.oo00)
= &/T 1.08111 + 0.409775 + 0.634313t2 - 0.093324E3 (4)
(SE = 0.7931 X fg(5)
r = l.oo00)
+
= tg/T = 0.581618 + 0.8757265 3.64626t2 1.35143t3 0.1739165‘ (5)
(SE = 0.23407
X
+
lO-l, r = l.oo00)
for 0.707 I 5 I3.0. SE designates the standard error of estimate and r is a multiple correlation coefficient. Equations 2-5 can be rewritten as
(3’) (4’)
Parameter Estimation
(5’)
The second-order-plus-dead-timemodel can be represented as G(s) =
K exd-es) Ps2+ 2T5s + 1
For a given value of € in the above second-order system with B = 0, one can estimate the value of t/T (say t l / T ) when the step response attains 10% of ita final value. It was also found by Huang and Clementa (1982) that the ratios, tl/T, t3/T, ..., are functions of C: only, and the range 0.707 < 5 C 1 could also be applied to a nonoscillatory process. In this investigation, values of t l / T ,t 3 / T ,t 6 / T , and t9/Tfor the range of 0.707 I 5 I 3.0 have been calculated by the golden-section method. Using a leastsquares method, these data are fitted by the following equations: fl(5)
= t l / T = 0.45465 + 0.060335 + 0.0167452 (2)
(SE = 0.83854 X
r = l.OOO0)
0888-5885/93/2632-0228$04.00/0
for 0.707 I I 3.0, where X 1 , X 3 , x6, or X 9 is the time at which step response attains 10%,30%,60%, or 90% of ita final value, respectively, and from the definition ti = X i - 0 for each i. On the other hand, let
The ratio a then becomes a function of 5 only. The inverse regreasion equation, giving [ as function of a,has also been obtained by a least-squares method as
+
= 7.40898 X
exp(16.3329~~) 100a 1.79015 X 10-2a3 4.55048a + 1.57083 2.25401 X 10d2a2- 1.14789~~ - 16.007 (6)
+
+
r = 0.99999) (SE = 3.4616 X which has a usable range 2.005 I a 4 5.508 (0.707 I 5 I 3.0). 0 1993 American Chemical Society
Ind. Eng. Chem. Res., Vol. 32,No. 1, 1993 229 Table I. Comparison of Parameter Values Obtained by the Least-Squares Method and the Present Method for Example 1 method K T E e SE least squares 1.0 1.9585 0.9374 0.8196 0.0044 present 1.0 1.9914 0.9134 0.8044 0.0058
Q)
m C
4Cxifi(D- Cfi(€)CXi 4Cfi2(€) - [Cfi(t)12
- - - - - - Approximaion
m
If t is obtained by eq 6,T and 0 can be solved by eqs 2'4 using a linear least-squares method as
T=
Actual res onse
0 Q 0.5
T = 1.9914
0.9134 0.8044
(7) 0.0
0.0
5.0 T ,;
- Cfi(t)CXifi(€) e = CXiCfi2(€) 4Cfi2(t)- [Cfi(t)12
-
15.0
10.0
I IIIIC
(8)
where
Cxi = x1 + x3 + x6 + xg Cfi(()= f I ( u + f 3 ( [ ) + f 6 ( 5 ) + f 9 ( t ) Zfi2(t)= f12(t)+ f a 2 ( ( ) + fs2(t) + f:(t) Cxifi(t)= X J1(t) + Xd3(t)+ Xde(t)+ X$g(€) Accordingly, a method to estimate parameters of the second-order-plus-dead-time model from step-response data has been developed. The following steps summarize the algorithm: 1. Calculate the steady-state gain K from the ratio of the output steady-state change to the input steady-state change (Coughanowr, 1991). 2. Obtain a = ( X , - X 6 ) / ( X 3- X , ) from step-response data. 3. Calculate t by eq 6. 4. Calculate T and 0 by eqs 7 and 8. Illustrative Examples Example 1. A fourth-order transfer function, 1 G(s) = (0.5s + l)(s + 1I2(2s + 1)
Figure 1. Comparison between real transfer function and model fit for example 1.
is assumed to be the actual process dynamics. From the actual step-response curve shown in Figure 1, one finds X I = 1.82,X3= 2.955,X, = 4.613,and X, = 7.798. Upon substitution of these values into the proposed algorithm, this transfer function is approximated by the second-order-plus-dead-time model. Figure 1 also indicates the comparison of the actual data with the model fitting suggested by the present method. In addition, a comparison between the least-squares fitting and the model fitting accomplished by the present method is given in Table I. Example 2. The transient response data of Brenner (1962)have been utilized to evaluate the effectiveness of the proposed technique and the second-order model for a wide range of nonoscillatory proc-. The data describe the relationship between the outlet and inlet concentrations of the axial dispersion model with Danckwerts' boundary conditions for various values of Peclet number (Pe). The Peclet number approaching zero is considered as a perfect mixing case or the first-order process, and the number approaching infinity is considered as a plug flow case or the pure time-delay process. Numerical results baaed on the present method and the leasbsquarea method (Marquardt, 1963) are given in Table 11. Number of iterations (NI) for each run using the present method as
Table 11. Comparison of Parameter Values Obtained by the Present Method and the Least-Squares Method for 0.2 I Pe I 10 present least squares run no. Pe T € e SE x 103 T E e SEX 103 NI 1 0.2 0.273 1.762 0.037 0.942 0.255 1.870 0.047 0.134 5 1.434 0.091 1.431 0.088 0.319 1.233 0.317 0.793 2 0.4 0.351 1.240 0.128 2.916 1.602 0.368 1.208 0.111 3 0.6 0.362 1.151 0.166 5.661 1.365 0.410 1.061 0.123 4 0.8 1.090 0.201 3.094 1.680 0.184 0.366 0.388 1.047 1.0 5 0.951 0.327 4.446 3.795 0.332 0.352 0.350 0.946 2.0 6 0.878 0.473 6.668 0.299 0.884 0.488 8.044 0.287 7 4.0 7.810 0.261 0.851 0.553 0.306 0.754 0.519 15.17 6.0 8 0.233 0.840 0.607 8.538 0.610 10.08 0.236 0.810 8.0 9 0.217 0.824 0.640 9.437 0.575 34.46 0.303 0.644" 10.0 10 "Outside the usable range.
Table 111. Comparison of Parameter Values Obtained by the Present Method and the Least-Squares Method for 10 I Pe I 160
present run no. 10 11 12 13 14
"E
= 0.8 is set.
Pe 10.0 20.0 40.0 80.0 160.0
T 0.223 0.158 0.111 0.080 0.062
least squares
P
e
SE x 103
T
c
e
SE x 103
0.8 0.8 0.8 0.8 0.8
0.631 0.741 0.819 0.872 0.898
10.848 14.667 14.574 12.735 14.220
0.217 0.160 0.115 0.082 0.056
0.824 0.809 0.806 0.800 0.810
0.640 0.739 0.814 0.868 0.910
9.437 13.453 13.303 12.310 9.775
NI 5 6 6 7 7
Ind. Eng. C h e m . Res. 1993,32, 230-235
230
an initial guess for the nonlinear least-squares fit is also given in Table 11. This comparative assessment reveals that the present method can give a reasonable approximation for Pe < 10. Difficulty arises for cases Pe 2 10 in estimating the parameters by the present method. This is due to very steep slope observed in the step-response curve for Pe 1 10, and the calculated €-value by eq 6 therefore falls below the usable range (0.707). The second-order model for t < 0.707is considered as a stronger underdamped case, and it is not suitable to fit a nonoscillatory process. However, by letting t = 0.8for the kind of case recommended by Huang and Clements (1982),T and can be solved by eqs 7 and 8. Table I11 shows the results. Further Comments A calculation technique to estimate the second-order parameters is presented using four points of step-response data. The main purpose of using four points is to form a ratio (a)which is function of 5 only. The four points suitably chosen as lo%, 30%,60%, and 90% of the final value in this study are considered to have a larger a range. Consequently, other similar four-point methods are also feasible. The damping ratio in the range of 0.707 I5 I 3.0 is used to fit the nonoscillatoryprocesses. Though the damping ratio in the range of 0.707I5 < 1.0 is described as an underdamped case, previous study (Huang and Clements, 1982)has recommended that this range could be used for nonoscillatory processes. This is also illustrated in the least-squares study for the large Peclet number in example 2. On the other hand, the a value is going to saturate for 6 > 3, and this may cause a sensitivity problem in the calculation of 6 using a. Thus, data for [ > 3 are not included in eq 6. To recapitulate, this technique will provide a good alternative for the rapid estimation of the second-order parameters from experimental data for control system design with a reasonable order of accuracy, or it will give a proper initial guess to further improve the fit for nonlinear least-squares regression.
SE = standard error of estimate t = time tl, t3,t6,t9 = time past the dead time at which step response attains lo%, 30%, 60%, or 90% of its final value, respectively T = time constant of the second-order model XI, Xs,X6,X9 = time at which step response attains lo%, 30%, 60%, or 90% of its final value, respectively a = (X,- X,)/(X, - XI) [ = damping ratio t9 =
dead time
Literature Cited Brenner, H. The Diffusion Model of Longitudinal Mixing in Beds of Finite Length. Numerical Values. Chem. Eng. Sci. 1962, 17, 229-243. Coughanowr, D. R. Process Systems Analysis and Control, 2nd ed.; McGraw-Hill New York, 1991; pp 296-299. Huang, C.-T.; Clements, W. C., Jr. Parameter Estimation for the Second-Order-Plus-Dead-Time Model. Ind. Eng. Chem. Process Des. Dev. 1982,21,601-603. Marquardt, D. W. An Algorithm for Least-Squares Estimation of Nonlinear Parameters. J. SOC.Ind. Appl. Math. 1963, 11, 431-441, Meyer, J. R.; Whitehouse, G. D.; Smith, C. L.; Murrill, P. W. Simplifying Process Response Approximation. Instrum. Control Syst. 1967, 40 (12), 76-79. Oldenbourg, R. C.; Sartorius, H. The Dynamics of Automatic Controls; The American Society of Mechanical Engineers: New York, 1948; pp 75-79. Seborg, D. E.; Edgar, T. F.; Mellichamp, D. A. Process Dynamics and Control; Wiley: New York, 1989; pp 173-176. Sten, J. W. Evaluating Second-Order Parameters. Instrum. Technol. 1970, 17 (91, 39-41. Sundaresan, K. R.; Prasad, C. C.; Krishnaswamy, P. R. Evaluating Parameters from Process Transients. Ind. Eng. Chem. Process Des. Dev. 1978, 17, 237-241.
* To whom correspondence should be addressed. Chi-TsungHuang* Department of Chemical Engineering Tunghai University Taichung 40704, Taiwan
Nomenclature
f(4) = function of damping ratio only G(s) = process transfer function
K = steady-state gain of the second-order model NI -- number of iterations Pe = Peclet number r = multiple correlation coefficient s = Laplace transform variable
Mao-Fa Huang Computer and Communication Research Laboratories Industrial Technology Research Institute Hsinchu 31015, Taiwan Received for review May 20, 1992 Revised manuscript received September 4, 1992 Accepted October 3, 1992
Intensification of Sorption Processes Using “Large-Pore”Materials Intensification of sorption processes by using “large-pore” adsorbents for linear and nonlinear isothermal casea haa been studied on the basis of the intraparticle diffusion/convection model. It is shown that the new design using “large-pore” packings is more reasonable than that obtained by reducing the particle size, particularly for processes operating in the laminar flow region. Intensification methods for sorption processes, such as adsorption, ion exchange, and chromatography, have been developed by Wankat (1987). The idea coming out from scaling rules is that using smaller packing size with higher mass-transfer rate and then the column length, column diameter, and operating time are scaled on the basis of the existing “old” configuration so that the productivity of the packing is largely improved while the separation performance and energy consumption are kept constant or improved in the Ynewnconfiguration. The original practice
of this idea is the use of rapid pressure swing adsorption (RPSA) processes (Jones and Keller, 1981),in which very small adsorbents and very short cycle times compared with conventionalPSA (Yang,1987)are used. The scaling rules were applied to linear and nonlinear elution chromatography by Wankat and Koo (1988),PSA processes by Rota and Wankat (19901,and size exclusion chromatography by Mohammad et al. (1992),respectively. However, intensification of an existing fixed-bed process by reducing the packing size will lead to a short and fat column in
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