Ind. Eng. Chem. Res. 1994,33, 174-176
174
Estimation of the Underdamped Second-Order Parameters from the System Transient Chi-Tsung Huang’ and Chin-Jui Chou Department of Chemical Engineering, Tunghai University, Taichung 40704, Taiwan, ROC
A simple calculation method was presented in this study for estimating the parameters of the underdamped second-order-plus-dead-time model from the system transient. Several estimation techniques without graphic aid or computer searching were recommended on the basis of the value of the maximum overshoot. The model parameters were estimated in the range of 0 < [ < 1using only a minimal number of data points along the step-response curve. This method was confirmed by the observed results as being both more reliable and easier to apply than currently available approaches for model parameter estimates.
Introduction Chemical processes in the open loop do not usually respond oscillatorily. However, obtaining an underdamped model would sometimes be desirable such as in (i) the closed-loopstep response and (ii) the case of cascade control system with the primary loop cutoff. The dynamics of an oscillatory system may generally be simplified by the underdamped second-order-plus-dead-time model as
\
\
h
4
v
u, g
za
c, 0.9c,
------Crnl
m Q,
P=
for 0 < f < 1. Methods have been explored for obtaining the model parameters from the step-response curve without computer usage by many authors in previous literature (Meyeret al., 1967;Sundaresanet al., 1978;Chen, 1989; Seborg et al., 1989). A simple calculation method was also proposed by Huang and Huang (1993) for estimating the nonoscillatory process dynamics using eq 1. However, combining the advantages of previous methods (Seborg et al., 1989; Chen, 1989; Huang and Huang, 1993) was the primary aim of this Research Note so as to form a simple and reliable way for estimation of the model parameters in the range of 0 < f < 1from the step-response curve. This technique could be expediently implemented in a pocket calculator or equally well in a digital computer.
Parameter Estimation Parameters of the underdamped second-order can be estimated from the extreme points of the step-response curve. A typical step-response curve for underdamped cases is shown in Figure 1. Chen (1989) modified the technique of Yuwana and Seborg (1982) using five data from three extreme points, i.e., Cpl, Cml, Cp2, tpl, and tml in Figure 1, for estimation of the model parameters. The advantage of this technique lies in its capability of obtaining accurate model parameters if the system is quite oscillatory. However, if Cmlcloses to C , and/or Cp2closes to C,, using this method (Chen, 1989) to estimate f may possibly result in a significant numerical error. Also, all of these extremes of the step-response curve may generally not be found in practical cases. The overshoots and undershoots for the underdamped second-order system can be obtained by the following equation (Kuo, 1991):
* To whom correspondence should be addressed.
Time Figure 1. Typical underdamped step-response curve.
overshoot or undershoot = (-1I-l
exp(-naf/(l-
t)’/2)
(2) where n = 1, 2, 3, etc. The third extreme value ( n = 3) is indicated from eq 2 to be indefinite for f > 0.4. Such approaches (Yuwana and Seborg, 1982; Chen, 1989) may therefore be concluded as being unsuitable for application toward cases of f > 0.4. However, values of f in the range of 0.4-0.8 are often used for specifying a desired control system response (Seborg et al., 1989; Ogata, 1990). A calculation method has recently been recommended by Huang and Huang (1993) for estimating the secondorder parameters of the nonoscillatory process transient using four points of step-response data. Extension of this approach (Huangand Huang, 1993)for underdamped cases is next delineated. Let t’ be the actual time (t) past the dead time; Le., ti’ = ti - 0 for each i. For a given 4 in eq 1without the dead time (e), the dimensionless time t‘/T (say tl’lT) can be estimated when the step response before the rise time attains 10% of ita final value. Values of tl‘lT, tS‘/T, t6’lT, and tdlT for the range of 0.4 If I1.0 are calculated in this investigation. Values of t[/T are only functions of f since the dead time (8) has been shifted out. These sets of data are then excellently fitted by a least-squares method as f,([)
t,’ t , - e = - = -- 0.451465 + 0.066696f T T
-
+ 0.013639f2
SE = 0.008 X lo9, r = LOO0 0 1994 American Chemical Society
(3)
Ind. Eng. Chem. Res., Vol. 33, No. 1, 1994 175
t i t,-e f3(0 T =T - 0.800879 + 0.1945505 + 0.101784[2
(4)
SE = 0.097 X f6({)
t i
t6-e
T
T
r = 1.000
= - = -- 1.202664 + 0.288331{
+ 0.530572t2 (5)
SE = 0.629 X lo-,, r = 1.000 t i t,-e f,([) = ?; = -= 1.941112 - 1.2372355
T
SE = 6.795 X
+ 3.182373t2 (6)
r = 1.000
for 0.4 5 4 5 1.0. SE designates the standard error of estimate, and r is a multiple correlation coefficient. If one lets (7) The ratio 8 then becomes a function of 5 only. The inverse regression equation, providing 5 as function of 8 for the range of 0.7 55 I1.0 (Le., 2.005 I8 I3.302), has also been obtained by a least-squares method as
5 = -0.460805 + 0.9763158 - 0.2545178’
+ 0.02811583 (8)
SE = 0.145 X
parameters of the underdamped second-order system for various ranges of the damping ratio. The reason for this inaccuracy is probably that the rising time is short for a more oscillatory system. In addition, values of tl, t 3 , t 6 , and t 9 are close to one another and cause numerical errors in the estimation of 5 in eqs 7 and 8. Also, the shortcoming of the four-point method is the requirement to interpolate four time points on the curve which is changing very rapidly. The damping ratio ([), however, can be easily estimated using the maximum overshoot (M,) of the stepresponse curve as (Seborg et al., 1989)
r = 1.000
Thus, the damping ratio (5) can be estimated from eqs 7 and 8 by using four points ( t l , t 3 , t 6 , and t g ) of the stepresponse curve. Also, if 5 is obtained by any technique, T and 0 can be solved by eqs 3-6 using a linear leastsquares method as
and
and
e = t, - ~ f ~ ~ t ) (12) From the authors’ experience, estimating 5 for a more oscillatory system is not an accurate approach despite the fact that the present four-point method can estimate
whereM, = (Cpl- C m ) / C mThe . extreme point Cpl is easily identified and requires no interpolation for more oscillatory cases. However, it is evidenced from eq 2 that the first peak (C,1) is indefinite for the sluggish response (say 5 > 0.8). For sluggish cases (say 4 > 0.81, the model parameters can only be estimated by eqs 7-10 using the four points (i.e,, t l , t 3 , t 6 , and t g ) . Thus, eq 8 is correlated in this study only in the range of 0.7 5 5 I 1.0. For cases in the range 0.4 I4 I 0.8,5 can be estimated by eq 13 using Mp, and estimating T and 0 either by eqs 9 and 10 using t l , t 3 , t 6 , and t g or else simply by eqs 11and 12 using tl and te. On the other hand, Chen’s method (1989) can be applied toward estimating model parameters for quite oscillatory cases, say f < 0.4. Judging from the above methods, the availability of 5 is suggested as being the prerequisite for implementing model parameters estimation. Once 5 is sought, T and 8 are then obtained by various techniques based on the available 5. 5 is a function of Mp only as expressed in eq 13,and Mpcould be directly obtained from step-resonance data. Also, Mp can be employed as an index for various cases of 5. Accordingly, a systematic technique based on Mpas an index is proposed in the following for estimating the model parameters for 0 < f < 1.0 from step-response data: 1. Calculate the gain (K)by dividing the steady-state output change by the input change (Seborg et al., 1989). 2. Calculate Mp, where Mp = (C,1 - C,)/Cm. 3. If Cpl is indefinite, say Mp < 1.5% (Le., 5 > 0.8) or sobthemodel parameters (E, T, and e) are then estimated by eqs 7-10 using the four-point method (Le., using t l , t 3 , t 6 , and t 9 ) . 4. If 1.5% IM, I25% (i.e., 0.8 I E I 0.4),the model parameters can be estimated by either of the following methods: (i) The five-point method (i.e., using Cp;,t l , t 3 , t 6 , and ts). Calculate 5 by eq 13, and then determine T and 8 by eqs 9 and 10. (ii) The three-point method (i.e., using Cpl,t l , and ts). Calculate 5 by eq 13, and then find T and 8 by eqs 11and 12. 5. If Mp> 25% (i.e., 5 < 0.4), which is a well oscillatory case, parameters estimation based on three extreme points (Chen, 1989) is recommended. In case the third extreme point (C,2) is indefinite, one can estimate 5 by eq 13 and then obtain T and 8 using Chen’s (1989) method.
Illustrative Example A typical cascade configuration for the inner loop, which was considered by Sundaresan et al. (1978), is selected as an illustration. The actual response C ( t ) to a unit step change in set point is plotted in Figure 2 as a solid curve. From the response curve, one has the relevant information
176 Ind. Eng. Chem. Res., Vol. 33, No. 1, 1994 14
Nomenclature
I
I
j
1.2
I - Actual response ...___ Approximation
,/
‘0.41
0.0
i
0.0
/
l
10.0
5.0
15.0
20.0
Time Figure 2. Comparison between actual response and model fit. Table 1. Comparison of Parameter Values Obtained by Various Methods method least squares Sundaresan et al. (1978) Chen (1989p proposed (five-point method) proposed (three-point method)
~e 1.063 1.211 0.966 1.122 1.124
E 0.685 0.566 1.248 0.612 0.591
0.604 0.588 0.519 0.596 0.596
SEX lo2 0.632 1.480 4.834 0.783 0.817
’Cpl = 1.097,Cml and
tml
0.983,Cpz = 1.003,C, = 1.O00,tpl = 4.797, = 8.346are used.
as t, =
1.150; t, = 1.696; t, = 2.384
t, = 3.215;
C,, = 1.097; Mp = 9.735%
Upon substitution of these data into the proposed fivepoint method, this system is approximated by the underdamped second-order-plus-dead-time model. The resultant curve with numerical data of the approximation is also shown in Figure 2. Alternatively, the model parameters have also been obtained by the proposed threepoint method simply using Cpl, tl, and t g . A comparison includingthe nonlinear least-squaresfit (Marquardt, 1963), the method of Sundaresan et al. (1976),the method of Chen (19891,and the proposed methods is listed in Table 1. The proposed methods are obviously comparable in efficacy to the nonlinear least-squares fit as indicated from an evaluation of the standard error of estimate (SE)in Table 1. The proposed methods are also confirmed from this table as being more reliable and easier to implement than others (Sundaresan et al., 1976;Chen, 1989). Incidentally, it appears from Table 1 that the proposed fivepoint method yields better results than the proposed threepoint method in light of the fact that more data points and more rigorous treatment are involved.
C(s) = controlled variable Cml = first minimum of system response C,1 = first peak of system response C,Z = second peak of system response C, = final system response f ( 4 ) = function of damping ratio only G(s) = system transfer function K = steady-state gain of the second-order model Mp = maximum overshoot = (C,1- C,)/C, r = multiple correlation coefficient s = Laplace transform variable SE = standard error of estimate t = time t’ = time past the dead time = t - 0 tl, t 3 , t 6 , tg = time at which step response before the rise time attains lo%, 30%, 60%, or 90% of its final value,
respectively tml = first minimum time of system response t,l = first peak time of system response T = time constant of the second-order model P = (t9 - t 6 ) / ( t 3 - tl) 0 = dead time = damping ratio
Literature Cited Chen, C.-L. A Simple Method for On-Line Identification and Controller Tuning. AZChE J. 1989,35,2037-2039. Huang, C.-T.; Huang, M.-F. Estimation of the Second-Order Parameters from the Process Transient by Simple Calculation. Znd. Eng. Chem. Res. 1993,32,228-230. Kuo, B.C. Automated Control Systems, 6th ed.; Prentice-Hall, Inc.: Englewood Cliffs, NJ, 1991;p 334. Marquart, D. W. An Algorithm for Least-Squares Estimation of the Nonlinear Parameters. J. SOC.Znd. Appl. Math. 1963,11,431441. Meyer, J. R.; Whitehouse, G. D.; Smith, C. L.; Murrill, P. W. Simplifying Process Response Approximation. Znstrum. Control S p t . 1967,40 (12),76-79. Ogata, K. Modern Control Engineering, 2nd ed.; Prentice-Hall, Inc.: Englewood Cliffs, NJ, 1990;p 266. Seborg, D. E.; Edgar, T. F.; Mellichamp, D. A. Process Dynamics and Control; Wiley: New York, 1989;pp 115-120. Sundaresan, K. R.; Prasad, C. C.; Krishnaswamy, P. R. Evaluating Parameters from Process Transients. Znd. Eng. Chem. Process Des. Dev. 1978,17,237-241. Yuwana, M.; Seborg, D. E. A New Method for On-Line Controller Tuning. AIChE J. 1982,28,434-439.
Received for review May 3, 1993 Revised manuscript received October 8, 1993 Accepted October 20, 1993.
Abstract published in Advance ACS Abstracts, December 1, 1993.
